Spaces of analytic functions on inductive/projective ... · and projective limits and results on...

Spaces of analytic functions on inductive/projective limits ofHilbert spacesCitation for published version (APA):Martens, F. J. L. (1988). Spaces of analytic functions on inductive/projective limits of Hilbert spaces. TechnischeUniversiteit Eindhoven. https://doi.org/10.6100/IR292007

DOI:10.6100/IR292007

Document status and date:Published: 01/01/1988

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https://doi.org/10.6100/IR292007

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https://research.tue.nl/en/publications/spaces-of-analytic-functions-on-inductiveprojective-limits-of-hilbert-spaces(5169b0bd-59f4-4d3c-9d1e-c1a6acfd09f3).html

SPACES OF ANALYTIC FUNCTIONS ON

INDUCTIVE/PROJECTIVE LIMITS OF

HILBERT SPACES

F.J.L. MARTENS

SPACES OF ANALYTIC FUNCTIONS ON

INDUCTIVE/PROJECTIVE LIMITS OF

HILBERT SPACES

PROEFSCHRIFf

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE TECHNISCHE UNIVERSITEIT EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. IR. M. TELS, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN

OP DINSDAG 8 NOVEMBER 1988 TE 14.00 UUR

DOOR

FRANCISCUS JOHANNES LUDOVICUS MARTENS

GEBOREN TE LITH

druk: wibro dîssertatiedrukker1j. helmond

Dit proefschrift is goedgekeurd door de promotoren:

Prof. dr.ir. J. de Graaf en Prof. dr. S.Th.M. Ackermans

Copromotor: Dr.ir. S.J.L. van Eijndhoven

Aan mijn ouders

CONTENTS

General introduction

I Preliminaries

§ 1. Locally convex spaces

§ 2. Inductive and projective limits

§ 3. Sequences and sequence sets

§ 4. Analytic functions

II Functional Hilbert spaces

Introduction

i

6

9

14

27

27

§ 1. Reproducing kemel theory 29

§ 2. Synnnetric Fock spaces as functional Hilbert spaces 44

§ 3. Examples of functional Hilbert spaces 55

Appendix 71

III Inductive and projective limits of semi-inner product spaces 77

Introduction

§ 1. Positive sequence sets

§ 2. Hilbertian dual systeins of inductive limits and projective limits

§ 3. Cross synnnetric moulding sets

Appendix

IV Spaces of analytic functions on sequence spaces

Introduction

§ 1. Generating sets

77

79

85

100

103

109

109

112

§ 2. Sequence spaces 120

§ 3. A sequence space representation of A(qi(]))) and A(w(]))) 127

§ 4. Elementary spaces in A(qi(]))) 134

§ 5. Compound spaces in A(q>(])))

Appendix

141

153

References 161

Index 164

Index of symbols 166

Summary 168

Samenvatting 169

Curriculum vitae 170

GENERAL INTRODUCT.ION

For a complex Hilbert space X, the so-called symmetrie Fock space

F (X) is defined as the direct sum of n-fold symmetrie Rilbert tensor sym

products of X. It is a topological completion of the symmetrie tensor

algebra of X. This Fock space, also called exponential Hilbert space, is

frequently used in quantum field theory by theoretical physicists and

can be represented as a functionalHilbert space S(X) with reproducing

kemel K(x,y) = exp(x,y)X, x,y € X. The Hilbert space S(X) consists of

analytic functions F on X with growth behaviour

2 IF(x) 1 ~ llFll8 (X) exp(illxllx) , x E X •

In a distributional approach of quantum field theory a symmetrie Fock

space construction has to be developed for locally convex spaces V,, e.g. test function spaces or distribution spaces, more general than

Hilbert spaces. In this general setting, it is not clear in which way

the symmetrie tensor algebra of V can be topologized or completed, a

difficulty that arises already at the level of the n-fold symmetrie

algebraic tensor products.

In the present monograph, these topological problems are solved rather

elegantly, constructing a 'symmetrie Fock space' of V which consists of

analytic functions on the (strong) dual V' of V. In this context we

observe that also the functional Hilbert space S(X 1) represents the

symmetrie Fock space F (X). sym

This thesis deals with a class of locally convex spaces, which admit

such symmetrie Fock space construction leading to locally convex spaces

of the same class. Our class consists of spaces which are inductive

limits of Hilbert spaces, projective limits of semi-inner product spaces

or both at once. The construction is carried out only for sequence

spaces and as a result we obtain locally convex spaces of analytic func

tions on sequence spaces. However, the concepts can be easily'.adapted

for more general locally convex spaces.

ii

The undèrlying treatise is based on an amalgamation of ideas from

- Aronszajn's reproducing kernel theory,

Bangmann's construction of Hilbert spaces of analytic functions,

which are repiesentations of the symmetrie Fock spaces F (ll:q) and sym F sym (R,2). general theory of analytic functions on locally convex spaces as

described by Nachbin and Dineen,

discussions on inductive/projective limits of (semi-) inner product

spaces by Van Eijndhoven, pe Graaf and Kruszynski. 1

Let us sketch our approach with an example. The spaces we start from in

this example are the so-called analyticity spaces and trajectory spaces

introduced in [G]. Spaces of these types be long to our class and have

been a major source of inspiration.

For a positive self-adjoint operator A in X, the continuous chain of

Hilbert spaces Xt, t E JR, is defined as follows. For t ~ O, Xt is the

space e-tA(X) endowed with the inner product (x,y)t = (etAx, etAy)X.

For t < O, Xt is the completion of X with respect to the inner product

(x,y)t = (éAx, etAy)X, There is a natural duality between the spaces

Xt and X_t. The set {Xt 1 t > O} is an inductive system with the ana

lyticity space SX,A as its inductive limit. The set {Xt 1 t < O} is a

projective system witb the trajectory space TX,A as its projective

limit. The spaces SX,A and TX,A are in strong duality.

In [EG 2] the spaces SFs (X),H and TFs (X),H are introduced for Ha suitable self-adjoint opr:ator associat~ with A. These spaces are

topological completions of the symmetrie tensor algebra related to SX,A and TX A' respectively .

• Besides, the analyticity space SFs (X),H and the trajectory space

TFsym(X),H admit a representation: spaces of analytic functions.

:_ndeed, in [Ma 1 ] the s paces SS (X), H and T S (X) , H are in troduced, where H denotes a positive self-adjoint operator in S(X) related to A. It bas

been proved tbat the space SS(X),H consists of analytic functions on

the strong dual TX,A of SX,A and tbat the space Ts(X),H consists of

analytic functions on the strong dual SX,A of TX,A'

iii

In this. thesis we replace SX,A and TX,A by more general inductive limits

V and projective limits W, which are in duality, and we construct spaces

of analytic functions on Wand V, respectively, similar to SS(X),H and

TS(X) ,H·

In the remaining part of this introduction we present a summary of each

chapter.

Chapter I contains prerequisites of the present monograph. We recall

results on locally convex spaces, in particular, results on inductive

and projective limits and results on analytic functions on locally

convex spaces. As an illustration serves the space of analytic functions

on the space of finite sequences. This function space is an important

tool in the aforementioned constructions.

Chapter II is devoted to Aronszajn's theory on reproducing kemels and

functional Hilbert spaces, We summarize results of Aronszajn's :theory

and represent the symmetrie Fock space F (X) as a functional Hilbert sym

space S(X). In a separate section we pay attention to examples of

functional Hilbert spaces, which are interesting for their own sake.

These examples are not relevant for the following chapters.

Chapter III is an exposition on inductive limits of Hilbert spaces and

projective limits of semi-inner product spaces. It contains a reformula

tion and a simplification of the theory as developed by Van Eijndhoven,

De Graaf and Kruszyóski. Starting from a Köthe sequence set and a

countable collection of Hilbert spaces, we construct an inductive limit

of Hilbert spaces and a projective limit of semi-inner product spaces

and describe their duality. Topological prope.rties of these inductive/

projective limits are put in correspondence with properties of the Köthe

set and the collection of Hilbert spaces .• In particular, certain Köthe

sets result in locally convex spaces, which are both inductive limits

of Hilbert spaces and projective limits of Hilbert spaces.

Chapters II and III contain the main tools for the final chapter, Chap

ter IV, and can be read independently.

In Chapter IV, for Köthe sets g, we introduce our sequence spaces

G. d[g] and G .[a], which fit in the description of Chapter III. The i.n proJ "'

iv

inductive limits Gind [g] arise from weighted R.2-spaces R.2 [a;ID], fixed

by nonnegative sequences ä on countable sets ]). The projective limits + +

G • [g] arise from semi-inner product spaces R.2 [a;ID]. Here R.2 [a;ID] prOJ denotes the Köthe dual of R.2 [a;ID].

Applying reproducing kemel theory, for nonnegative sequences a on 1D

we introduce our elementary function spaces F. d[a] and F .[a]. The in proJ functions in Find [a] are analytic on R.;[a;1D] whereas the functions in

F .[a] are analytic on R.2 [a;ID]. Fora Köthe set g the collection prOJ {F. d[a] 1 a E g} turns out to be an inductive system and the collection in {F .[a] 1 a € g} a projective system. Our compound spaces F. d[g] and prOJ in F .[g] are defined to be the corresponding inductive and projective prOJ limits, The space F. d[g] consists of analytic functions on G .[g] and i.n · proJ yields a description of the symmetrie Fock space of G. d[g], Likewise, in the space F .[g] is a space of analytic functions on G. d[g] and . proJ in yields a description of the symmetrie Fock space of G .[g]. proJ

We charac terize the func tions in F • . d [g] and F . [g] wi th a growth in proJ

estimate and we characterize them also in terms of the coefficients in

their monomial expansions. The latter characterization leads to sequence

spaces G. d[up[g]] and G .[up[g)], homeomorphic to F. d[g] and in proJ i.n F .[g], respectively, with up[g] a Köthe set on multi-indices. prOJ Studying these sequence spaces, we obtain topological properties of the

corresponding compound spaces.

As an illustration of the theory we represent suitable infinite dimen

sional Heisenberg groups as groups of linear hcmeomorphisms on the

spaces F. d[g] and F • [g]. rn proJ

CHAPTER 1

PRELIMINARIES

In this chapter we present the prerequisites of the entire monograph.

§ 1. Locally convex spaces

In this section we present definitions and results on locally convex

spaces. Sources are the well-known textbooks [Yo], [Con] and [Sch].

(1.1) A aomplex veator spaae is a set V with two algebraic operations,

addition and scalar multiplication, which satisfy:

(V,+) is an abelian group.

- The scalar multiplication is a mapping from C x V into V for which

for all À,µ € C and v,w € V

À(v+w) • Àv + Àw

(À+ µ)v Àv + µv

(Àµ)v À(µv)

1v = v •

In general, we omit the adjective 'complex' in the expression 'a com

plex vector space'.

A t(lpological veator spaae is a vector space V endowed with a topology,

such that the mappings

(v,w)i+v+w, (v,w) E V x V

and

(À,v) 1+ Àv , (À,v) E C x V

2 I Preliminaries

are continuous with respect to the product topology. Hence for an open

set W in a topological vector space V the sets ÀW and v + W are open for

all À E t and v E V.

(1.2) A set W in a vector space V is

aonve;i; if \J Vv,wEW vÀ,0<À<1 Àv + (1-/,)w E W

balan,..ed if .., u " vvEW vÀ€(:, 1À1 ;l;l ÀV € W

absorbing if VvE V 3À>O : Àv E W •

(1.3) A locally aonvex apaae is a Hausdorff topological vector space,

such that the topology is locaUy aonve~, i.e. each neighbourhood of o

contains a convex, balanced and open set. We abbreviate 'locally convex

space' and 'locally convex topology' by 'L.C. space' and 'L.C. topology'.

A semino:rrn p on a vector space V is a function p: V + IR+, such that for

all À € t and v,w € V

p(v + w) :i p(v) + p(w) p( v)

1.1. Lemma

Let V be a (topological) vector space. Let W be a convex, balanced and

absorbing (open) subset of V. The Minkowsky functional or gauge kw• defined by

1 -1 Itw(v) = inf {À > 0 À v € W} , v € v

is a (continuous) seminorm on V.

1.2. Lemma ..

Let V be a (topological) vector space. Let p be a (continuous) seminorm

on V.

The set Up= {v € V j p(v) < 1} is a convex, balanced and absorbing

(open) subset of V.

The previous lemmas are the key to the main result in this section,

namely the relation between L.C. topologies and collections of seminorms.

I Preliminaries

(1.4) A collection of seminorms Pr is

{py j y E r} on a vector space V

sepaPating if for each v E V, v f. o, there exists y E r such

that py(v) f. 0 ,

di~eated if for each y1,y2 € r there exists y Er such that for

all v E V: p (v) ::î p (v) and p (v) ::î py(v). Y1 y Yz

(1.5) All subsets Win V, satisfying the condition:

For each w E W there exist .a finite set E c: r and s > 0 such that

w + n su c: w, yEE Py

establish a topology, T(V,pr)·

Let T denote a topology in a vector space V. The collection of semi

norms Pr on V gene~ates T if T = T(V,pr>·

Here follows the main result in this section.

1 . 3. Theorem

Let V denote a vector space endowed with a topology T.

3

Then V is a L.C. space iff T is generated by a separating collection of

seminorms.

Let Pr and qfi denote two collections of seminorms on a vector space V. The collections Pr and qfi are equivalent if Pr and qó generate the same

topology; to state it differently, the seminorms Py~ y E r, are contin

uous with respect to the topology T(V,qfi) and the seminorms q6, o E ó,

are continuous with respect to the topology T(V,pr).

The following result is useful.

1. 4. Lemma

Let Pr and qfi denote two directed collections of seminorms on a vector

space V. Then Pr and qö are equivalent iff

4 I Preliminaries

and

(1.6) A subset W of a L.C. space V is bounded if for each neighbour

hood U of o there exists À > 0 such that À-1w c U.

1. s. Lemma

Let W denote a subset of a L.C. space V. Then the set W is bounded if f for all continuous seminonns p on V sup {p(v) 1 v E W} < 00 •

In a L.C. space all compact subsets are closed and bounded.

The space of all continuous linear operators from a L.C. space V into a

L.C. space W will be denoted by B(V,W). We write B(V) instead of B(V,V).

1.6. Lemma

Let V and W denote L.C. spaces with generating directed collections of

seminorms Pr on V and q/:J. on W, respectively. Let A denote a linear

operator from V into W. Then A E B(V,W) iff

1. 7. Corollary

Let V and W denote L.C. spaces. Let B be a bounded subset of V and let

A E B(V ,W). Then A(B) is a bounded subset in W.

(1.7) The du.at of a L.C. space Vis the vector space of all continuous

linear functionals on V and is denoted by V'.

For each bounded subset W of V we define the seminorm -OW on V' by

-Ow(R.) = sup { 1R.(w)1 l w E W} , R. E V' •

I Preliminaries 5

The ûJeak dual of a L.C. space V is the space Vf endowed with the topol

ogy T(V', {-OW 1 Wis a finite subset of V}).

The weak dual of V is denoted by V&·

The strong dual of a L.C. space V is the space V' endowed with the

topology T(V', {-OW 1 W is a bounded subset of V}).

By VS we denote the strong dual of V.

Weak duals and strong duals of L.C. spaces are L.C. spaces.

(1. 8} For each v E V we define the linear functional .ev on V' by v

ev (i) = i(v) , v i E V'

The L.C. space V is reflexive if the mapping v i+ evv is a homeomorphism

from V onto (V$)e•

(1.9) A Cauchy net (vi)iEJ in a L.C. space V ie. 'a mapping j i+ vj from

a directed set J into the space V such that for each neighbourhood W of

o in V:

3.EJ V.EJ '<" v. - v. E W • l J 'l.~J l J

A Cauchy net (vi)iEJ in V has a limit v E V if for each neighbourhood W of o:

v-v.EW. J

The L.C. space V is aonrplete if each Cauchy net has a limit.

(1.10) The L.C. space Vis semi-MonteZ if each closed and bounded subset

. of V is compact. The semi-Montel space v· is MonteZ if V is reflexive.

( 1. 11) A c losed, convex, absorbing and balanced subset W of a L.C. space

V is a barrel. The L.C. space V is barreZed if each barrel in V is a

neighbourhood of o in V.

(1.12) The L.C. space V is bornologicaZ if every convex balanced subsèt

of V, which absorbs every bounded set in V, is a neighbourhood of o.

6 I Preliminaries

(1.13) A linear operator S from a Banach space V into a Banach space W is called nuaZear if there exist bounded sequences (v~)nEIN in VS and

(wn)nEIN in W and a sequence C € t 1 (IN) with nonnegative entries such

that

S(u) = l c(n)v' (u)w , nElN n n

u € v .

If V and W are Hilbert spaces, then the operator S is nuclear iff S is

a trace class operator.

Let V denote a vector space, p a seminorm on V and Np" { u € V 1 p( u) " 0}.

The normed linear space V is the quotient space V/Np with norm p de-"' p À

fined by p(v + N ) " p(v), v E V. By V we denote a completion of Vp and p À p we consider Vp as a subspace of Vp. Let q denote another seminorm on V

with VvEV : p(v) :îi q(v),

The.n VvEV: v + N cv+ N and p(v+N):;; q(v+N ). q ~ - p q

The canonical mapping jq,p: Vq +Vp is the uniquely determined continu-

ous mapping, which satisfies jq,p(v+ Nq) = v + Np, v € V.

A L.C. space V is nuclear if for each continuous seminorm p there exists

a continuous seminorm q such that

VvEV : p(v) :;; q(v)

and A A

the canonical mapping j V + V is nuclear. p,q q p

§ 2. Inductive and projective limits

In Chapter III we study dual pairs of L.C. spaces consisting of an

inductive li111it and of_ a projective limit. These inductive limits and

projective limits originate from directed collections of (semi-) inner

product spaces.

In this section we mention some generalities on inductive/projective

limits. For further references see [Sch], Chapter II, Sections 5 and 6

and [Con], Chapter IV, Section 5.

(2.1) Let V and W denote two topological vector spaces with V c W. The canonicaZ injection j from V into Wis defined by j(v) = v, v E V.

I Prelim.inaries

If the canonical injection is continuous, we express this by writing

V c;.. W.

2. 1. Deflnition

Let A denote a directed index set.

A collection of vector spaces· {Va ! o: E A}, each endowed with a L.C.

topology, is an inductive system if for each a,6 € A, with o: $ S,

Vac;..Ve.

2. 2. Deflnition

Let {Va 1 a € A} denote an inductive system,

7

The inductive limit, lim ind V , is the vector space U V,., endowed with o: E A 0: aEA "

the finest L.C. topology, such that Va c U V for each 13 E A. This "" aEA a

topology is called the inductive limit topology.

A neighbourhood basis B of o in lim ind V is formed by all convex o o: E A o:

balanced set U c U V~ such that for all ex E A the set U n V is open o:EA" · a in va. A criterium for continuity of mappings on inductive limits reads:

2. 3. Theorem

Let {Va 1 a E A} denote an inductive system. Let F denote a linear map

ping (seminorm) from lim ind V into a topological vector space (IJ (into o: E A

.IR+).

Then F is continuous iff for each S E A the linear mapping (seminorm)

FIVa is continuous on v8•

Proof:

Cf. [Con], Section IV.5.

The second part of this section deals with projective limits.

2.11. Deflnitioh

Let A denote a directed index set.

A collection of vector spaces {Va 1 o: E A}, each endowed with a L.C.

topology, is a p~ojective system if for each o:,S E A, with a $ $,

Vac;..Va.

c

8

2.5. Definition

Let {Va 1 a E A} denote a projective system.

The projeative Zimit, lim proj V , is the vector space

I Preliminaries

n v endowed a€A a a E A et

with the coarsest topology, such that n v c;_ va etEA et

for each 8 E A. This

topology is called the projeative limit topoZogy.

We remark that for the

of v E n V is given etEA a

projective limit topology a neigbbourhood basis

by all intersections fl F 1 (U ) , where U,.... is a aEE a a "'

neighbourhood of v in va and where E is a f inite subset of A.

The next results are not very surprising.

2.6. Lemma

Let {Vet 1 a E A} denote a projective system where each Va is endowed

with topology T(Va•Pra). Let r • a~A ra and let Pr = {py restricted to fl V j y E r}. Then

a.EA a

a. The topology of lim proj V,,,,, is equal to T( n V,...., Pr). aEA "' aEA....,

b. The space lim proj Va is a L.C. space iff the collection Pr is , et E A ·

separating.

Proof:

See [Sch], Chapter II, Section 5.

2. 7. Theorem

Bet {Va. 1 a E A} denote a projective system. Let F denote a linear map

ping from a topological vector space W into lim proj V • a. E A a

Then F is continuous iff f or each a. E A the mapping F is continuous

from W into V ó'.•

Proof:

See [Sch], Chapter II, Section 5.

As is well known, eacb complete L.C. space is a projective limit of

Banach spaces.

In the next section we give two classica! examples, one of an inductive

limit and one of a projective limit. See Lemma.3.t.

From Lemma 2.6 it follows that not every projective limit is Hausdorff.

In general, it is hard to check whether an inductive limit is Hausdorff.

IJ

IJ

i' ;1

I Preliminaries

§ 3. Sequences and sequence sets

Throughout this section, Il denotes a fixed, countable set. Functions

from Il into t will be called sequences (labeled by Il) and tbey are

denoted by a,b, "., etc. By w(ll) we denote the set of all sequences

labeled by ll.

For each subset JE of U the sequence Xm is defined by

j e: JE

j t E

So Xm is the characteristic function of the subset E. In particular, we errploy the notation n = Xn• 0 = x0.

Fix u € w(Il). The suppor-t of u, which we denote by ll[U], is the set

{j € Il i u(j) r O}. lts complement is denoted by llc[u], viz.

Xc(U] = {j E U ! U(j) = O} •

Correspondingly we have the functions nu and Ot1•' defined by

and

Hence for all j e: l[

We mention the following operations on sequences:

Let u, v E w(lI), À e: t:.

Add.ition. The sequence u+ v € w(X) is defined by

[u + v](j) = u(j) + v(j) , j e: Il •

Saa'lar multipliaation. The sequence ÀU e: w(X) is defined by

[ÀU](j) = À[U(j)) , je:x.

Pl'odwt. The sequence u • v € w(lI) is defined by

[u • v](j) = u(j)v(j)., j e: u •

9

10 I Preliminaries

Thus w(ll) becomes a commutative algebra.

Tensor product. Let w E w(Il) with JI a countable set. The sequence

u © w E w(n: x JI) is def ined by

u 0 w(i,j) = u(i)w(j) , (i,j) E ][ x JI •

Absolute value. The sequence lul E w(JI) is defined by

lul(j) = lu(j)I j E ll •

Pseudo-inverse. The sequence u-1 € w(ll) is defined by

l [u( ') 1-1

-1 J u (j) =

0 jEll.

. . -1 The sequence u is called the pseudo-inverse of u.

-1 -1 We remark that u • u = u • u = 11 u.

Let p, o denote subsets of w(JI) and let T denote a subset of w(n:).

We employ the following notations:

u • p = {u • r r E p} •

p•o•{r•s rEp,sEo},

pl!h={r©t rEp,tET}, etc.

We introduce the following linear subspaces of w(JI):

(3.1) The space of finite sequences q:>(lI), defined by

q:>(lI) =.{v € w(l[) 1 #(lI:[v]) < oo} •

(3.2) The space of null sequences c0

(:n:), defined by

c0 (1I) = {vE w(l[) i Ve:>O: #({j EU 1 IV(j)t > e:}) <co}.

(3.3) The Banach space of all bounded sequences ~00(ll), defined by

~00(]() = {v € w(U) I sup({ lv(j) 1 1 j € ](}) < co}

with norm lvl00

= sup({V(j) I j € n:}).

l

i\ Il I'

I Prelil!linaries

(3. 4) For p € 1N the Banach spaces R,p (lI), defined by

R, (][) - {v € w(lI) 1 I lv(j) 1P < <XI} p jElI

Evidently, the space t 2(lI) is a Hilbert space and we denote the inner

product by (•,•)2

•

We now endow the spaces q>(Il) and w(lr) with suitable L.C. topologies.

To this end we introduce the following notion:

(3.5) A sequence (lrq)qElN is called an e~haU8tion of lI, if it satis*

fies:

u lI =lI q€1N q

Each enumeration of ][ induces an exhaustion.

Assume that we have an exhaustion. Let q E lN.

We define the seminorm p on w(lI) by p (X) = 1 Xu • x 12

• q q q

The q-dimensional Hilbert space q> (lI) is the space x • q>(:O:) with q liq

norm x 1+ lxlI • xl 2 • q

11

The topological vector space w (:0:) is the space w(lI) with the topology q

generated by the seminorm p • q

The collect ion {<Pq (:0:) 1 q E JN} is an inductive system and

u <P (:0:) = (fl(lI) • q€1N q

The collection {w (lI.) 1 q € lN} is a projective system and q

n w (lI) = w(:n:) • q€JN q

12 I Preliminaries

In the sequel, by ip(JI) and w(:n:) we mean the L.C. spaces lim ind <P (:n:) q EIN q

and lim proj w (lI), respectively. q E IN q

For u € w(lI) respectively ip(JI) we define the seminorm qu on ip(JI)

respectively w(lI) by

qu (x) = 1 x • u 11 , x E qi(JI) respectively w(Il) •

The next lemma establishes that the topologies on qi(:lI) and w(:n:) are

Hausdorff and do not depend on the particular choice of the exhaustion,

3.1. Lemma

a. The inductive limit topology on ip(lI) is equal to

T (i.p(lI), {qu 1 u E w(JI)}) •

b. The projective limit topology on w(E) is equal to

T(w(E), {qa 1 a € !.P (1[) }) •

Proof:

a, Let U E w(JI) and let q .E IN. Then

Hence the seminorm qui(.!) (Il) is continuous. By Theorem 2.3 it follows

that qu is a continuous qseminorm on tp(ll) = lim ind tp (ll).

Conversely, let p denote a continuous seminorin E ~ li: ind L p(X{·}·>lxO>I = qu(x). jEll J

Lenu:na 1.4 yields statement a.

b. By Lemma 2.6 the projective limit topology on w(lI) is equal to

T(w(JI), {pq 1 q € IN}). With the aid of Lemma 1.4 we can prove that

T(w(JI), {pq 1 q €IN}) = T(w(JI), {qa 1 a E tp(II)}). D

I Preli111inaries

The spaces tp(ll) and w(ll) are Montel spaces. This result is a conse

quence of the theory stated in Chapter III as is the case with the

following characterizations.

(3.6) Let W denote a subset of q>(ll).

The set W is compact iff there exists q € IN such that W c: q> (:U:) q

and W is compact in q>q (ll).

(3.7) Let W denote a subset of w(ll).

The set W is compact iff for each j € 1I the set {u(j) 1 u E W} is

compact in t.

We introduce the set of multi-indices labeled by E.

3. 2. Defini tion

Let n denote a countable set.

The set of multi-indices l1(Il) c:q>(Il) is the set of all sequence from

n: into IN+ (= 1N U {O}) with a finite number of non-zero entries, i.e.

l1(JI) = {s: lI + lN+ 1 #(ll [s]) < oo} •

We call l1(ll.) the multi-index set labe'led by IL

For n E IN+ we also define

(3.8) For elements of l1(ll) we introduce the following corresponding

multi-index notations.

For each s E l1(ll) and a E w(Il):

s 1 = n s (j) 1 • jEll

a5 = n [a(j)]s(j) • jEJI [s]

In particular, we have a~ = 1.

We remark that 11({1, ••• ,q}) = m1.

13

14 I Preliminaries

§ 4. Analytic functions

In this section we introduce the concept of analytic function on an

open subset W of any L.C. space according to S. Dineen [Di 1] and

present some of its consequences. We study two examples: The space of

analytic functions on q>(ID) and the space of analytic functions on w (ID).

(ID is a countable set.) But we first have to deal with the special case

of analytic functions of a finite number of complex variables.

Following L. Hörmander in [Hö], we introduce the notion of analytic

function on an open subset in ~q by means of the Cauchy-Riemann equa

tions.

Let W denote an open set in ~q. By Ck(W) we mean the space of k times

continuously differentiable complex valued funétions on W where

k = oo,1,2, •••• To each function F € c1(W) we associate the differential

form dF of the function F.

To this end, we identify fq with 1R2q, viz.

z = x + iy '

Let F € C 1

(W). Considering F as a function of 2q real variables it

k . ()F d óF 1

,,. • < ma es sense to wnte ax(j) an äYITT , " J = q. The expression

q óF ~ oF l ax(3•) dx(j) + t.. ay(3') dy(j) j=1 j=1

is called the differentiaZ fo1'm of F and is denoted dy dF.

Put

dF 1 êF i óF äZGî ~ 2 axGT ~ 2 äYITT , dz(j) • dx(j) + idy(j)

and

dZ(j) = dX(j) - idy(j) •

Then

dF = ! a:CJ•) dZ(j) + ! ~ dz(j) , j=1 j•1 az (j)

I Preliminaries

With

q and aF = I ~ dï'îJ)

j=1 azur we have dF = ()F + ä'F •

4. 1. Definition

A function F E c1(W) is analytic if äF • O on W, i.e. F satisfies the

Cauchy-Riemann equations on W

By A(W) we denote the vector space of all analytic functions on W.

First we present some results related to Càuchy integrals.

For u E 4:q and r E IRq with r(j) > 0 we denote the polydisc

lw(j)- u(j) 1 ~ r(j), 1 ~ j ~ q}

by n B(u,r), its interior by n B(u,r), .and its distinguished boundary q q

by n aB(u,r). q

lw(j) - u(j) 1 = r(j)}

4.2. Lemma ([Hö), Theorem 2.2.1)

15

Let F be a continuous complex valued functión on an open subset W C:: tq.

Then the funct!on Fis analytic on W iff for all polydiscs nqB(u,r) c:W and all V E n B(u,r)

q

F(V) (27Ti) -q

n élB(u,r) q

Here we use the multi-index. notation:

q (w-v) 11 = n (w(j)-v(j)) 1

j=1

See also (3. 8).

and d W = dW(1) ". dW(q) • q

16

4. 3. Corollary ([Hö], Corollary 2.2.2)

Let F E A(W).

I Preliminaries

Then the function F belongs to c"'(W) and all derivatives of F are ele

ments of A(W).

Let F E A(W). By a.F we mean the partial derivative of F, corresponding

h ·th . blJ . " (lp 1 to t e J varia e, 1.e. "j F = oZ(j) , ;;;; j ;;;; q.

Fors E IN~ we denote the partial derivative [ ri ()~(j)] F by a5 F. j=1 J

4. 4. Corollary

Let F E A(W) and let s € IN;. Q

Let n B(u,r) c W. q

Then for all V E TI B(u,r) q

4. 5. Corollary

f

Let F E A(W) and let nqB(u,r) cW. There exists µ > 0 such that for all

F(W)/(w -v) 5+ 11 dw. q

As in the one dimensional case, analytic functions of several variables

have power series expansions.

4.6. Theorem ([Hö], Theorem 2.2.6)

Let F denote a complex values function on an open set W c~q. Then F belongs to A(W) iff for all nqB(u,r) cW and all v € TiqB(u,r)

F(V) l ft (()5 FJ(U) • (V - u) 5

, sEm;

where the series converges absolutely and unifonnly in v on n B(u,r). <!

The power series in Theorem 4.6 can be replaced by a series of'homo

geneous polynom.ials.

I Preliminaries

4. 7. Definition

Let F E A(W).

Then for each u E Wand n E lN+ we.define the n-homogeneous polynomial

F [u], associated tó F, by n

v € t:q •

4. 8. Definition

Let F denote a complex valued function on an open subset W c: Çq.

Then F is ray-analytic on W if for each u E W and v € t:q the function

À + F(u + ÀV) is analytic in a neighbourhood of 0 in C.

4.9. Lemma

Let F € A(W).

Then F is ray-analytic on W and for each U € W and V € Cq there exists

a neighbourhood U of 0 in 4: such that

00

F(u+ ÀV) • l Àn Fn[u](v) , n-0

À€ u

where the series is absolutely convergent in À on U.

The following lemma is a consequence of Lemma 4.2.

4.10. Lemma

Let F denote a continuous and ray-analytic function on an open subset W in (:q.

Then F is analytic on W.

The continuity condition in the previous lemma is superfluous •. Indeed,

Hartogs' theorem yields the converse of Lemma 4.9. So it yields a

remarkable characterization of analytic functions in q variables.

4.11. Theorem {Hartogs' theorem, [HöJ, Theorem 2.2.8)

Let F denote a ray-analytic function on an open subset W in (:q.

Then the function F is analytic on W.

17

18 I Preliminaries

Next we extend the notion of analyticity to functions defined on an

open subset of a L.C. space.

4. 12. Definition

Let F denote a complex valued function on an open subset W of a L.C.

space V.

The function F is ray-analytia on W if for all u E W and v E V the

function À 1+ F(u+ ÀV) is analytic in a neighbourhood of 0 in c. The function F is analytia on W if F is continuous on W and ray-analytic

on W. By A(W) we denote the space of all analytic functions on W.

It follows from Hartogs 1 theorem that a function F is analytic on a

L.C. space V iff F is continuous on V and ray~analytic on each finite

dimensional subspace of V.

For the following results we refer to Dineen [Di 1], Chapter 2, Sections

1 and 2.

First we introduce n-homogeneous functions associated to a ray-analytic

function. An n-homogeneous function G on a vector space V is a function

which satisfies

G(ÀV) = Àn G(v) • ÀEC, VEV.

4. 1 3. Deflnition

Let F denote a ray-analytic function on an open subset Win a,L.C.

space V. For u E W and n E lN+ we define the n-homogeneous function Fn[u]: V-+ C by

F [u]Cv) • lim -t--- J F(u+ ÀV)/Àn+1 dÀ • n p+O ~1

IÀl•p

v € v .

We give two characterizations of the concept of analyticity.

4.14. Lemma

Let F denote a ray-analytic function on an open subset W in a L.C.

space V. Then

a. The function F belongs to A(W) iff for each u E W the series

I Preliminaries

v 1+ Iri'=0 Fn[uJ(v) defines a continuous function in a neighbourhood

of o in V.

b. The function F belongs to A(W) iff F is locally bounded on W, i.e.

for each u E W there exists a neighbourhood Ou of u in W such that

F(Ou) is a bounded set.

Proof:

See [Di 1], Section 2.2.

19

We give an example, interesting in itself, but also important as a tool

in Chapter II. Let H denote a Hilbert space with inner product (•,•)H

and let y E H. The function F on H is defined by F(X) = exp(x,y)H,

x E H. The function F is analytic on H. The n-homogeneous functions .

Fn[u], u E H, are v t+ (v,y)n exp(u,y)H, v E H.

We now come to the two examples mentioned _in the introduction of this

chapter, viz. the analytic functions on q>(lD) and on w(lD).

First we deal with q>(lD). As we have seen, the vector spaoe q>(lD) is a

union of finite dimensional vector spaces with inductive limit topology.

We show that this topological property of qi(lD) implies that each ray

analytic function on q>(lD) is continuous on q:i(lD). Thus we arrive at an

extension of Hartogs' theorem, for which we present a detailed proof,

because we could not find a reference.

IJ. 1 s. Theo rem Let F be ray-analytic on i.p(lD).

Then F is analytic on q>(lD).

Proof: Let (ID q) qElN be an exhaustion of ID and let !Pq (ID) be the fini te dimen-

sional vector space xlD • q>(lD) with norm x 1+ lxlD • xl 2• Then q q

q>(ID) lim ind i.p (lD) q E lN q

Fix q E lN. Since tpq (lD) is a fini te dimensional vector space, Hartogs'

theorem implies that the restriction FllA (lD) is analytic on <Pq(ID).

Hence FjlA (ID) is continuous on \Pq(ID). ihe only failing argument in

the proofqis the continuity of F on 1P(lD). For this we prove the

a

20 I Preliminaries

remarkable fa.et that each function on q>(ID), continuous on every <Pq (ID),

is continuous on q>(ID) itself.

Indeed, let u E q>(ID) and let e: > O. Then u E IP (ID) for some p E IN. p

Hence there exists 6 Em., 0 < 6 < 1/2P, such that p p

"'xE (ID) : lx- u 11 < o • IF(x) - F(u) 1 < e:/zP • (j)p p

Let q >p. The set C = {v E (j) (ID) J lv-ul1

1} is compact in lP (ID). q q q

Hence there exists 6 , 0 < 6 < 1/2q, such that q q

"'xyEC :lx-y11<éi •IF(x)-F(y)l<r;,/2q. • q q

We define a E w(ID) by

~ 110 ,

a(j) = P

1 / éi ' q

. E ID J p

j E ID '-ID 1 , q > p q q-

and the continuous seminorm qa on q>(ID) by

Let v E l(lq (ID) for some q E lN with qa (v - U) < 1. We have

q IF(V) - F 1 :il r IF<v. xID ) - F(V. xID >1 +

k=p+1 k k-1

+ JF(v•xID )-F(u)I p

Since

1 q qa (v - U) = 0 J (V. - U)XID J 1 + l J V • XID , 10 11 < 1 ,

p p k=p+1 k k-1

for p < k :il q

1 v • x10 - u 11 < o p p

v • Xm E ck k

1 v • x10 - v • Xm 11 < ok • k k-1

I Preliminaries

Hence q

IF(v) - F(u)I ~ L E/2k + E/2P ~ s. k-p+1

4. 16. Corollary

Let F denote a complex valued function on q>(ID).

The function F belongs to A(ll)(ID)) iff for all finite dimensional sub

spaces W of q>(ID) the function Flw is analytic on W.

As all spaces A(V), the space A(!.l>(ID)) is an algebra, because for

F ,G € A(q>(ID)) the function F • G is analytic on q>(ID).

We need a pairing between q>(ID) and w(ID).

4.17. Definition

The pairing <•,-> between q:i(ID) and w(ID) is defined by

<x ,y> • }'. x(j >YGT • j€ID

where x € q>(ID), y € w(ID) or x € w(ID), y € q:i(ID).

We define the following operations on functions F in A(<,p(ID)).

(4.1) The translations Ta, a € q>(ID), defined by

[Ta F](x) = F(a+ x) , X € q>(ID) •

(4.2) The shifts Rb, b € w(ID), defined by

[Rb F](X) = e<x,b> F(X) • x € q:>(ID) •

(4.3) The dilatations 0b, b €.w(ID), defined by

(4.4)

[0b F](x) = F(b • X) , X € ip(ID) •

The multipliers Q., j €ID, defined by J

[Qj F] (X) = <x,e/. F(X) • x € Q)(ID) •

where e . = X{ • } , j € ID • J ' J

21

D

22 I Preliminaries

(4.5) The differential operators a., j E lD, defined by J

[o3. F](x) = lim [F(x + ;\e.) - F(x) ]/À ,

;\....O J x E (j)(lD)

(4.6) We recall that lM(lD) denotes the set of multi-indices labeled

by lD. Cf. Definition 3.2.

The differential operators 35, s E lM(lD), defined by

a5 F= n [a~(j)]F. jElD J

Theorem 4.15 and Corollary 4.16 imply that these operations are indeed

linear mappings from A(q:i(lD)) into A(<P(lD)).

As in Cq for each u E t:p(lD), each fini te subset 1F in lD and each sequence

r on lF with positive entries, we denote the polydisc

{w E <P(lD) l lD[w] clF, vjEJF: lw(j)- u(j) 1 ;:;;. r(j)}

by nF B(u,r). Its distinguished boundary

{w E tp(lD) l lD[w] c:JF, vjEJF: lw(j)-u(j)J r(j)}

is denoted by OF <lB(u,r).

We have the following Cauchy integral representation for the partial

derivatives.

4.18. Lemma

Let F E A(tp(lD)), u E q>(lD) and S E lM(lD).

For fini te subsets F of lD w.ith lD[u], lD[S] c F we have

(27Ti)- #(JF)

where dF w means n dW (j) . jElF

Proof:

f s+x

F(W)/(w - U) lF dF W

nF ClB(u,r)

The proof is based on Lemma 4.2 and Corollary 4.4 for a sufficiently

large fini te dimensional subspace of q>(ID). D

I ·preliminaries 23

As in the case of q c0111.plex variables, analytic functions on qi(lD) have

a power series expansion.

IJ. 19. Theorem

Let F E A(qi(lD)). Then

vxEqi(lD) : F(x) = }'. -fï [as F](o)xs SElM(lD)

where the series converges absolutely and uniformly in x on compact

sets in q>(lD) ~

Proof:

Let (ID ) ElN denote an exhaustion of lD and let q> (lD) = xID • <P(ID). q q q q

The function F\ (lD) is analytic and can be expanded in a power series. IPq .

Hence

"'xE (ID) : F(x) " l J... [as F] (o)xs • <Pq SE:M(ID),ID[S]cID SI

q


sets in q>q(ID). See Theorem 4.6.

The proof is canpleted because the canpact sets in q>(lD) are always

canpact sets in some q> (ID). Cf. (3. 6). c q

Although q>(ID) = U qi (ID), it is not true that each function qElN q

F € A(q>(lD)) depends only on a finite number of variables; so it is not

true that F = BxF F for some finite subset lF of ID. As an example we

mention the function G: X .+ <x,a>, x E <P(lD), where ID[a] is infinite.

The function G is analytic on q>(ID) and for each finite subset lF of ID

G F 9xF G.

We finish the discussion of A(IP(lD)) with some remarks on the transla

tions T a and the shifts Rb.

(4. 7) Let G{q>,w}(lD) denote the group q>(ID) x w(ID) x t with product

operation

For each appropriate triple (a,b,c) the linear operator

24 I Preliminaries

U(a,b,c): A(<P(ID))-i- A(q>(ID)) is defined by

, U(a,b,c)F é Rb T a F F E A(q>(ID))

We have

The group G{q>,w}(ID) can be regarded as a generalization of the Heisen

berg group. See the Appendix to Chapter IV. The mapping U is a represen

tation of the group G{q>,w}(ID) in A(q>(ID)).

The second example is established by the space of analytic functions on

the space w(ID). It turns out that these functions have a simple charac

terization.

4. 20. Theorem

Let F denote a ray-analytic function on w(ID).

The function F belongs to A(w(ID)) iff there exists a finite subset JF

of ID such that VX(w(ID): F(x) = F(xF• x).

Proof:

-. By Hartogs' theorem the restriction FIXlF •W(ID) is continuous and

also the operator x >+ XlF • x is continuous from w(ID) into XlF • w(ID).

Hence Fis continuous on w(ID).

•· Theorem 4.15 says that Flq>(ID) is analytic and Theorem 4.19 yields

v F( ) \' _!_, [as F](o)xs • XEq>(ID) : X SE:m.f(ID) S.

We prove that F depends· only on a finite nl.lllber of variables.

Since F is continuous, there exist a E q>(ID) and µ > O, such that

VWEw(ID): 1a. Wl1 < 1 " IF(W) 1 < µ •

Let s E 1M(ID). We define r E w+ (ID) by r = a + ptla for some p > o. With the integral representation, given in Lemma 4.18, we ,can derive

the following estimate:

I Preliminaries

Since p is arbitrary, we find that for all s € lM(ID) with

ID[s ]'-ID[a) 1' t', [as F](o) = o.

25

Put F = ID[a], Then we have Flqi(ID) = (0XF F)lqi(ID)' Since qi(ID) is dense in w(ID), we get F = e F. c

Xp

Also the space Á(w(ID)) is an algebra.

Further, we list some operations on the functions F in A(w(ID)}, These

operations are similar to the operatións in the previous example.

(4.8) The translations Ta• a E w(ID), defined by

[Ta F](x) = F(a+ x) , x € w(ID) •

(4.9) The shifts Rb, b E q>(ID), defined by

[Rb F](x) = e<x,b> F(X) • X € w(ID) •

(4.10) The dilatations eb, b E w(ID), defined by

ceb F](x) = F(b • x) , X € w(ID) •

(4.11) The multipliers ,Qj' j €ID, defined by

[Q. F](x) = <x,e.> F(x) , J J

X € w(ID) •

(4.12) The differential operators aj' j €ID, defined by

[a. F](x) = lim (F(X+Àe.) - F(X))/À, X € w(ID). J Ä-+0 J

(4.13) The differential operators as, S € lM(ID), defined by

as F ... [ n a~ (j >] F • jEID J

Theorem 4. 20 implies that these operations are linear mappings from

A(w(ID)) into A(w(ID)).

An analytic function on w(ID) also has a power series expansion.

26

4. 21. Theorem

Let F E A(w(ID)). Then

vxEw(ID) : F(x) = l -& [as F] (o)xs SElM(ID) .

I Preliminaries


sets in w(ID).

Proof:

Because of Theorem 4.20, there exists a finite set lF clD such that for

all x E w(ID)

F(x) = l 5\ [as F](o)x5

• SE:M(ID) ,ID[s ]cF

Let W denote a compact set in w(ID). From (3. 7) it follows that XF • W

is a compact set in the fini te dimensional vector space XlF' • W(lD).

Hence the last mentioned power series converges absolutely and uniformly

in X on W.

We finish this section with a remark on the 'Heisenberg group' related

with A(w(ID)), Cf. (4.7).

(4.14) Let G{w,q>}(ID) denote the group w(ID) x q>(ID) x (; with product

operation

For an appropriate triple (a,b,c) the operator

V(a,b,c) A(w(ID)) + A(w(ID))

is defined by

V(a,b,c)F = ec Rb Ta F , F E A(w(ID)) •

Similar to the case (4.7), V is a representation of G{w,q>}(ID) in A(w(ID)).

c

Introduction

CHAPTER ll

FUNCTIONAL HILBERT SPACES

Aronszajn's paper [Ar] on reproducing kemels, published in 1950, is

the starting point of this chapter.

A reproducing kemel reproduces a functional Hilbert space in the fol

lowing way: Let H denote a functional Hilbert space, i.e. a Hilbert

space consisting of complex valued functions on a set E such that for

all y €Ethe evaluation functionàls F '+ F(y), F € H, are continuous.

Their Riesz representants K , complex valued functions on E, constitute y

the reproducing kernel Kof H, defined by K(x,y) • K (x). Functions y

K: E x E -+ t, thus defined, are functions of positive type on E, viz.

VR.EJN VOl..€t,x.€E, 1;;i;jH J J

Conversely, starting with a function P of positive type on E, there

exists a uniquely determined functional Hilbert space H(E,P), which

admits P as its reproducing kernel.

We mention an important result of reproducing kemel theory: Given

functions K1 and K2 of positive type on E, we write K1 ~ K2 whenever

Ki - K1 is of positive type. If K1 ;;; K2 then H(E,K1) c::;..H(E,K2).

In Chapter IV we will consider inductive systems ·and inductive limits.

The inductive systems are composed of functional Hilbert spaces, con~

sisting of analytic functions on qi(])), the space of finite sequences.

We mention three points where reproducing kemel theory enters in the

considerations of Chapter IV:

First: Reproducing kemel theory provides an elegant and transparent

way to introduce functional Hilbert spaces of analytic functions on

q>(JD), which are in fact representations of symmetrie Fock spaces.

Cf. Section IV.4.

28 II Functional Hilbert spaces

Second: A directed set of functions of positive type on a set E corre

sponds to an inductive system of functional Hilbert spaces on E. Cf.

Theorem IV.4.4.

Third: The functions in a functional Hilbert space are characterized by

a growth estimate, which only involves the reproducing kemel. This

result carries over to inductive limits of functional Hilbert spaces.

Cf. Theorem IV.5.4.

After this explanation on the relevance of this chapter for Chapter IV,

we now sketch the contents of the present chapter.

In the first section we deal with functions of positive type. We char

acterize the functions in a functional Hilbert space and study a partial

ordering and some operations in sets of functiàns of positive type. It

turns out that Hilbert tensor products and direct sums of functional

Hilbert spaces are again functional Hilbert spaces.

The second section deals with the n-fold Hilbert tensor product Tn(H),

the n-fold synunetric Hilbert tensor products T5 ym(H) and the symmetrie n

Fock space F (H) of a Hilbert space H. Each of these spaces admits a sym functional Hilbert space as its representation. The representations of

T8 ym(H) and F (H), denoted by S (H) and S(H), consist of analytic n sym n functions on H. (We regard Sn(H) and S{H) as the n-fold symmetrie

Hilbert tensor product of H and the symmetrie Fock space of H, respec

tively.) Guichardet's monograph [Gu] on symmetrie Hilbert spaces touches

upon the same subject, but is based on a completely different approach.

The third section consists öf classical examples of functional Hilbert

spaces consisting of analytic and harmonie functions. Our spaces S(Cq)

appear to be the Bargmann spaces B of order q. q

In the appendLx we stud~ linear operators in the symmetrie Fock space

S{H), such as the annihilation and creation operators of quantum field

theory and differential operators of infinite order.

II Functional Hilbert spaces 29

§ 1. Reproducing kemel theory

For a greater part the results in this section are inspired by Arons

zajn' s results [Ar] on reproducing kemels. First, we introduce func

tional Hilbert spaces, reproducing kemels and functions of postive

type and we explain the mutual relations between .these notions.

1 • 1. Definition

Let H denote a Hilbert space consisting of complex valued functions on

a set E.

The space H is called a fu:national Hilber1; spaae if for each y E E the

evaluation functional F 1+ F(y), F E H, is continuous.

Throughout, the norm and the inner product in a Hilbert space ,ff will be

denoted by ll•llH and (•,•)H' respectively. For functions L: Ex E +(:we

write L for the function xi+ L(x,y). y

1. 2. Definitlon

Let H denote a Hilbert space consisting of complex valued function on

a set E.

A function K: E x E + (: is called the reprodwing kemel of H if K bas

the following two properties:

- Vy€E : The function Ky belongs to H.

An i111111ediate consequence of the Riesz 1 representation theorem is the

following

1.3. Lemma

Let H denote a Hilbert space of complex valued functions on a set E.

Then H is a functional Hilbert space iff H has a reproducing kemel K.

Reproducing kemels of functional Hilbert spaces are uniquely deter

mined.

Let K be the reproducing kemel of a functional Hilbert space H on E.

For all Jl. € IN and a. € (:, y. E E, J J

:lî j :lî R,, we have


According to the following definition K is a function of positive type.

1 , 4. Definition

Let P denote a function from E x E into ~.

The function P is called a funation of positive type on E if for all

~ E 1N and a. E C, y. E E, 1 ~ j ::! ~. J J

By PT(E) we denote the set of all functions of positive type on E.

Each reproducing kemel of a functional Hilbert space on E is a function

of positive type on E. The converse is also valid.

1. 5. Theorem ([Ar], Part 1, 2( 4))

Each element P of PT(E) induces a uniquely determined functional Hilbert

space on E, which admits P as its reproducing kemel.

Proof:

We give a sketch of the proof. In the linear space V(E,P) = <{Py 1 y E E}>

we introduce the inner product

Si nee

~ ~

1 L a: P (x) 1 ::! llP llV(E P) Il Î aJ. PyJ. llV(E,P) • j=1 J y j x , j•1

each Cauchy sequence in V(E,P) is pointwise convergent on E. Thus we

arrive at a completion of V(E,P) which consists of pointwise limits of

Cauchy sequences and which aèlmits P as its reproducing kemel'. D


1. 6. Deftnltion

Let K € PT{E).

The space H(E,K) is defined to be the functional Hilbert space induced

by K.

We make some remarks.

(1.1) Let K € PT(E). For all x,y € E we have that

K(x,y) • K(y,x) ,

2 llKyllH(E,K) • K(y,y) ;:: 0 ,

jK(x,y)i 2 ~ K(x,x)K(y,y)

(1.2) The pointwise product of two functions of positive type on Eis

again a function of positive type. Cf. Lemma 1. 17.

31

(1.3) Each finite dimensional Hilbert space of functions on a set Eis

a functional Hilbert space.

Summarizing: Each function of positive type on E is a reproducing kemel

of precisely one functional Hilbert. space on E.

The elements of a functional Hilbert space admit the following simple

characterization, which can be found in a monograph of T Ando. Cf.

[And], Chapter II, Theorem 1.1. The characterization is a growth esti

mate condition.

1. 7. Lemma

Let K € PT(E) and let F denote a complex valued function on E.

Then the function F belongs to H(E,K) iff there exists y > 0 such that

for all i € lN and o.. € C, y. € E, 1 ~ j ~ i, J J

Proof:

•. Let i € lN and a. € 4:, y. € E, 1 ~ j ~ i. Then we have J J


1 Î a.. F (y.) 12

j=1 J J

t 2

l( I a..K ,F) 1:;; j=1 J y j H(E,K)

""· On the linear space V(E,K) we define the linear functional m by

t R, m( I a.. K ) = I a. F(y.) •

j=1 J yj j=l J J

The assumption implies that m is well defined and that m is bounded

on V(E,K). Therefore m extends to a continuous linear functional on

H(E,K) with Riesz representant G. So, for all y E E

Hence, F =GE H(E,K).

1. 8. Corollary

Let F E H(E,K). Then

2 llFllH(E,K)

= sup { \ Î a.. "F<Y:T\ 2( Î <ik a 3• K(yk,yJ. >)-1

1 i Î aJ. 1),3• llH(E,K) F o} .

j=1 J J k,j•1 j=1

Proof:

The statement is equivalent to

llFllH(E,K) = sup { 1(G,F}H(E,K)1 1 G E V(E,K), llGllH(E,K) 1} •

Since V(E,K) is dense in H(E,K), the statement follows.

As already announced in the introduction to this chapter, we introduce

a partial ordering in PT(E).

1. 9. Definition

Let Kt•Kz € PT(E).

By K1 ::> Kz we mean K2 -K1 E PT(E). ([K2 -K1](x,y) = K2(x,y)-K

1(x,y).)

a

a


The next theorem is one of the most useful results in this section. It

shows how two functional Hilbert spaces fit in each other.

1.10. Theorem ([Ar], Part 1, 13, Corollary IV2)

Let K1 and K:z belong to PT(E).

a. If K1 :;; yK2 for some y > 0, then H{E,K1) c;. H{E,K2) and

Proof:

We put (•,•)i and ll•Ui for (•,•)H(E,K.) and D•DH(E,K.)" 1 l.

a. Let K1 :;; yK2• Then for F € H(E,K1) we have the estimate

Lemma 1.7 implies that F € H(E,K2) and Corollary 1:a implies that

llFll2

:;> ./Y llFll1

•

33

K € V(E,K1

). we define 1,yj

itence G • 0 impli~s VFEH(E,Ki) : (F ,G ) 2 == 0 and G F 0 implies

G* F o. . For GE V(E,K1), G F O, we define the linear functional !G on H(E,K1)

by !G(F) = (F,G) 1/llG*ll 2, F € H(E,K1).

For all G E V(E,K1), G F O, we have

HGHI • llGH/RG*llZ ,

VF€H(E,K1) : ltG(F)I :iî llFll2 •


The Banach-Steinhaus theorem implies that there exists a > 0 such

that VGEV(E,K1): llGll 1 ::; ollG*ll 2, whence for all fl E IN and aj E ~.

yj E E, 1 ~ j ~ fl,

Hence K1

The proof of Theorem 1.10 bas been proved with the aid of Lemma 1.7.

The proof in [Ar] is based on different arguments. However, Theorem 1.10

is also an extension of Lemma 1.7.

(1.4) Let F denote a complex valued function on E. The function

~ E PT(E) is defined by ~(x,y) = F(y)F(x), x,y E E.

Then H(E,~) is a one-dimensional Hilbert space with orthonormal basis

{F} and

1 r a. F<Nl2

j=1 J J

A reformulation of Lemma 1.7 in terms of Theorem 1.10 reads: F E H(E,K)

l.• ff 3 " K y>O:~•Y·

The next lemmas clarify the relations between orthonormal systems in

H(E,K) and. the reproducing kernel K.

1.11. Lemma

Let (4>n)nEI denote an orthonormal system in H(E,K) for some K E PT(E).

Then

a. For all x,y E E the series lnEI 4>n (y) 4>n (x) is absolutely convergent.

b. The function K41 : Ex E ~ t defined by Kq,(x,y) = InEI 4>n(y)4>n~x)

belongs to PT(E) and Kq, ~ K.

c. The system (4>n)nEI is an orthonormal basis iff Kq, =K.

Cl


1.12. Lemma Let P denote an ortbogonal projection in H(E,K) and let the function

L: E x E -+ ~ be defined by

L(x,y) = [P(Ky)](x) , x,y € E •

a. Then L belongs to PT(E) and L ~ K.

b. If P(H(E,K)) is endowed with the Hilbert space structure induced by

H(E,K), then P(H(E,K)) = H(E,L).

Proof: We only prove part b. For all F € P(H(E,K)) and y € E.we have

Hence, P(H(E,K)) has reproducing kemel L.

The functions of positive type, introduced in the two previous lennnas,

are of the same kind.

1 . 1 3. Lemma

Let K,L € PT(E) such that H(E,L) c H(E,K).

The following statements are equivalent:

a. There exists an orthonormal system (~n)n€I in H(E,K) such that

V L(x,y) = l ~~ (x} • x,y€E n€I n n

b. VF€H(E,L) : llFllH(E,L) = llFllH(E,K) •

c. There exists an orthogonal projection P in H(E,K) such that

Vx,y€E : L(x,y) = [P(Ky)](x) •

Proof:

a. "b. Let G € V(E,L). Then G = If.1 aj LYj and

a

36

\' ä. ex.. (L ,L )H(E K) k..t=1 K J y j Yk ' .


2 llGllH(E,K)

Since V(E,L) is dense in H(E,L), statement b. is valid.

b. •c. By asstnnption H(E,L) is closed in H(E,K). Let P denote the

orthogonal projection from H(E,K) onto H(E,L). For all y E E and

F E H(E,L) we have

whence L = P(K ) • y y

c.=>a. Let (<l>) EI denote an orthonormal basis in the range1of P. Let n n 2

y E E. Then Ly = P(Ky) = ÎnEI an <l>n with ÏnEI ja.nl < 00•

For all m E I we have

Hence for all x,y E E: L(x,y) = Ï EI ~<l> (x). n n n

Now we study operations on functions of positive type. We start with

restriations of functions of positive type.

1. 14. Lemma ([Ar], Part I, 5, Theorem 1)

Let K E PT(E) and let E1

c E.

Consider the restriction K1 = KIE xE . Then 1 1

F E H(E,K).

for some

llGllH(E K ) =min {llFllH(E,K) 1 F E H(E,K), G = FI E } ; 1 • 1 1

The following results deal with algebraic operations on functions of

positive type.


1.15. Lemma ([Ar], Part 1, 6)

Let K1 ,K2 E PT(E) and let K1 + ~ be defined as usual by (K1 + ~) (x,y)

= K1(x,y) + K

2(x,y). Then

. 2 llFllH(E K +K ) "'

• 1 2

37

As a preparation of the next lemma we introduce Hilbert tensor products.

For j = 1, ••• ,n let H. denote a separable Hilbert space with orthononnal J ... n

basis (e. ,e)iEID • Let H = X. 1 H •• J, j J= J

By T(H1, •••• Hn) we denote the Hilbert tensor product of H1

, ••• ,Hn. The

elements of T(H1, ••• ,Hn) are continuous n-linear functions F: H ~ C with the additional property that

l F(e1 . , ••• ,e . )G(e1

. , ••• ,e . ) · EID · EID •l1 n,Jn •l1 n,Jn 3 1 p••"Jn n

The inner product does not depend on the particular choice of the

respective orthonormal bases in Hj. The inner product space T(H1, ••• ,Hn)

is a Hilbert space. (In fact we have T(H1, ••• ,Hn) identified with a

representation of it.)

For y E H we define y(1) © • " © y(n) E T(H1, •• "Hn) by

(y(1) 8 ", 0 y(n)](X) n n

j•1 (x (j) ,y (j)) H. " x E H

J


.... For u,v E H we have

(u{1) © ••• © u(n), v(1) © ••• @ v(n))T(u H) n1 t • • • ' n

The set

n

{e1 ,k{1) © ••• © en,k(n) 1 k E X mj} j•1

is an orthonormal basis in T(H1 , ••• ,Hn).

n n (v{j),u{j))H .•

j=l J

Now, let Hj = H(Ej,Kj)' 1 :;; j n. Put Ê = Xj.1 Ej'

For each F E T(H1

, ••• ,Hn) we define the complex valued

on Ê by

function Ev [F] n

.... For Fj E H(Ej,Kj)' 1 ~ j :;; n, and x € E we have

1.16. Lemma ([Ar], Part I, 8)

n n F. (x(j)) •

j=1 J

Let K. E PT(E.), 1 :;; j:;; n, and let the function©~ 1

K.: Ex Ê ~ C be J J J= J

defined by

n n © K.(x,y) "' n K.(x(j),y(j)) ,

j=1 J j=1 J .

Then

a. n 0 K. E PT(Ê).

j=1 J

.... x,y E E

b. The mapping Evn is an antilinear isometry from T(H(E1,K1), ••• ,H(En,Kn)) .... n ) ontoH(E,©j=l Kj

The last algebraic operations are the pointwise multiplications of func

tions of positive type.


1.17. Lemma ([Ar], Part 1, 5, Tlleorem Il)

Let K1 •Kz € PT(E). The product K1 • Kz: E x E + t is defined by

[K1 • K2 ](x,y) = K1 (x,y)K2 (x,y). x,y € E. Then

b. A function G: E + t belongs to H(E,K1

• K2) iff there exists

F € H(E x E, K1@K2) sucb that VxEE : G(x) = F(x,x) •

Proof:

We have the relation: K1(x,y)K2(x,y) = [K1 @ Kzl((x,x),(y,y)).

Put E1 = {(x,x) 1 x E E}. Identifying x and (x,x) we have the relation

39

Kl. K2 = K1@ KzlE XE • Apply Lemma 1.14. D 1 1

The last operations on functions of positive type we study, are impor

tant for the description of the symmetrie Fock spaces S(H) of a Hilbert

space H. See Section 2 of this chapter.

1. 1 8. Definition

Let K.E PT(E).

The function exp[K]: Ex E + t is defined by

exp[K](x,y) = exp(K(x,y)) , x,y E E •

1.19. Lemma

Let K € PT(E).

Then exp[K] € PT(E).

Proof:

Let R, € 1N and ctj € t, yj E E, 1 ~ j ~ t. Lemma 1.17 implies that


Exponentiation preserves the ordering of functions of posi~ive type.

First we give an auxiliary result, which will be used frequently in

Chapter IV.

1. 20. Lemma

Let K,L,M E PT(E) with K L.

Then K • M :;; L • M.

Proof:

Lemma 1. 17 implies that L • M - K • M

K • M :;; L •M.

1 • 21. Corollary

Let K,L E PT(E) with K :ii L. Then

b. exp[K] :;; exp(L] •

Proof:

(L - K) • M be longs to PT (E), whence

a. We prove this statement by induction. First K L. Secondly, assume

that t11 :;; Lm. Twofold application of Lemma 1.20 yields t11+ 1 Lm • K:;; :;; Lm+l •

b. This statement is a consequence of statement a.

The following theorem deals with monotone sequences in PT(E).

1 • 22. Theorem

Let (Lq) qEIN be a sequence in PT(E) and let K E PT (E) such that for all

q E IN Lq :;; Lq+l ;;_; K. Then

a. The sequence (Lq (x.y)} qElN is convergent for all x,y E E •.

b. The function L: Ex E + è defined by L(x.y) = lim L (x.y), x,y E E, q-+oo q

belongs to PT(E) and satisfies VqEIN: Lq :ii L :;; K.

c. For all y E E lim llL - L . llH(E L) q-+oo y q,y • 0.

d. The space U H(E,L ) is dense in H(E,L). qElN q

D

0


Proof: We start with a claim:

{1.5) Let M1,M2 € PT(E) with M1 ;:; M2• Then

Proof of the claim:

Let y €E. By Theorem 1.10 the function Ml;y belongs to H(E,M2) and

So

a. Fix x,y € E. Let p,q € lN with q ii:: p. Then by (1.5) we have

:li (L (y,y)-L (y,y))L (x,x). q p ~

Since for all u € E the sequence (Lk{u,u))kElN is a monotone non

decreasing sequence with upperbound K(u,u), the sequence (Lq(x,y))qElN

is convergent.

b. Let Jl E lN, aj E C:, yj E E, 1 ::> j :;; Jl. For all q,p E lN we have

Jl Jl

I ~CLj Lq(yk,yj) ;:; I (ik CLj Lq+p (yk,y j} ;:; k,j=1 k,j=1

R. :i l (ik CLj K(yk,yj) •

k,j=1

whence in the limit p ~ oo it follows that L E PT(E) and for all

q E lN Lq :i L :li K.

c. Fix y E E. From property (1.5) it follows that


From statement b. it follows that (Lq;y)qElN is a Cauchy sequence in

H(E,L) with limit Ly.

d. Let F € H(E,L) such that F .L H(E,L ) for each q E lN. For all y E E q

we have

F(y) = lim (F ,L • )H(E L) = 0 , q-+<» q ,y '

whence F 0 • 0

We conclude this section with some simple results concerning representa

tion theory.

Let G be a group with identity element e. The Banach algebra of bounded

operators on a Hilbert space H will be denoted by B(H) and the identity

on H by I.

1. 23, Definition

Let G be a group, let H be a Hilbert space and let 1T denote a mapping

from G into B(H).

The mapping 1T is called a representation of the group G in H if it

satisfies

(1.6) vr,s€G 1T(rs) n(r)'!T(s) ,

(1.7) 1T(e) = I

The representation 1T is a unitary representation if

( 1. 8) -1

VrEG : (n(r)) = 1T(r ) •

Let G be a topological group.

The representation .1T is a aontinuoua representation if

(1.9) VhEH : .The mapping r l+ n(r)h, r E G, is continuous at e.

Let V be a subspace of the Hilbert space Hand let A E BCH). If

A(V) c V, then the subspace Vis called A-invariant. If also A*(V) c V, then V is said to reduae A. If A is a unitary operator, then each A

invariant closed subspace also reduces A.


1. 2 ll. Definition

Let ff denote a unitary representation of a topological group G in H. The representation n is an il'reilui:tible representation if each closed

subspace V of H, which is n(r)-invariant for all r € G, equals either

{o} or H.

Let K € PT(E). A bijection from E onto E is called a tranefonnation on

E. Under certain conditions on a group G consisting of transformations

on E, the group G admits a unitary continuous representation in H(E,K).

1. 25. Definition

Let K € PT(E) and let R denote a transformation on E.

The complex valued functi.on Il(R)F on E is defined by

[Il(R)F) (x) = F(R-1 x) , x € E •

1. 26. Theo rem

Let K € PT(E) and let R denote a transformation on E.

Then Il(R) is a unitary operator on H(E,K) iff

Vx,y€E : K(x,y) = K(Rx,Ry) •

Proof: -1 -1 -1 "°'• Fix y €E. Since Il(R ) = [Il(R)] we observe that Il(R )KRy € H(E,K).

So for all F € H(E,K) we have

(F-,Il(R-l )~y)H(E,K) = (Il(R)F, Il(R)Il(R-l )~y)H(E,K)

"' (II(R)F, ~y)H(E,K) " F(R-1

(Ry)) = F(y) •

Now uniqueness of the reproducing kemel K in H(E,K) yields

K(Rx,Ry) = [Il(R-t)~y](x) = K(x,y) •

•. Fix F € H(E,K). Because of Lemma 1.7 the function IT(R)F belongs to

H(E,K) and Corollary 1.8 yields

llII(R)FIH(E,K) = llFllH(E,K) • 0


1. 27. Corollary

Let K E PT(E) and let G denote a group of transformations on E.

Then the mapping IT is a unitary representation of G in H(E,K) iff

V GV : K(x,y) = K(Rx,Ry) • RE x,yEE

In addition, if G is a topological group, then the representation IT is

continuous if f

Proof:

Indeed, the mapping IT satisfies conditions (1.6) to (1.8) in Definition

1.23. Let G be also a topological group.

"'°'• Let y E E. The mapping Ri+ IT(R)K, REG, is continuous at the point y

IE. Since IT(R)Ky KRy' we get.

•. From the assumption it fellows that for all G E V(E,K)

lim llII (R)G - GllH(E,K) == 0 • R+IE

A simple argument based on the triangle inequality yields that for

all F E H(E,K)

lim llII(R)F- FllH(E,K) "' 0 • R+IE

§ 2. Symmetrie Fock spaces as functional Hilbert spaces

In this section symmetrie Fock spaces will be described as functional

Hilbert spaces. They play an essential part in the theory of analytic

funetions spaces in Chapter IV. Symmetrie Foek spaees appear in the

description of quantum field tbeory. They are direct sums of n-fold

symmetrie Hilbert tensor products. For the preeise definition see (2.7).

0


We start with the n-fold (symmetrie) Hilbert tensor products.

Throughout this section H denotes a separable Hilbert space with ortho

nonnal basis (ej)jElD. The inner product (•,•)H is a function of two

variables, also be denoted by Q.

(2.1) For n € JN, the n-foül Hitbert tensor product of H is the Hilbert

tensor product T(H, ••• ,H) and is denoted by T (H). n

Since Q € PT(H), Lemma 1.16 implies that Q©n = 9~ Q belong to PT(H°"). 1=1

2. 1. Theorem

Let n € JN.

Then T (H} "!' H(Hn,Q@n) •. n

Proof:

Let F € Tn(H) and U € ffl. Since F is a continuous n-linear function we

have

F(U)

• (F, u(1)0."9u(n))T (H) • n

So Tn(H) is a functional Hilbert space. lts reproducing kemel N satis

fies

N(x,y) = (y(1)9."0y(n), x(1)9".9x(n))T (ff) n

n n (x(j),y(j))H,

j=1

Note that H(H,Q) is a functional Hilbert space representation of H.

(2.2) Let G denote the group of linear transfonnations S: ff1 + ff1 . n 9n Q

such that for all x,y € ffl: Q (Sx,sy) • Q n(x,y).

0

Clearly, the representation 1T of G in T (H), defined by 1T (R)F = F 0 R-1 n n n n

is unitary.


First we discuss some special elements of G • n

(2.3) For unitary operators Rj on H, 1 :il j :il n, the operator

R1 @ • • • 0 Rn in Gn is defined by

(2.4)

by

Fora permutation cp of {1, ••• ,n} the operator cp EG is defined n

w(x) = (x(cp(l)),.",x(cp(n))) '

(2.5) For r E IRn with nj=l r(j) = 1, the operator Dr E Gn is defined

by

(r(1)X(1), ••• ,r(n)x(n)) , x € tf .

The elements of G admit the following characterization: n

2.2. Lemma

For all n E lN and all SE Gn there exist a permutation cp of {1, ••• ,n},

unitary operators Rj on H, 1 :il j :il n, and r E IRn with riJ=î r(j) = 1

such that S = <ÎI o [R1

@ • " © R ] 0 D • n r

Proof:

The proof is by induction. For n = 1 the statement is valid. Assume the

statement is valid for n =Jl. Let SE GQ,+1

• For convenience we denote . . ~1 •

the elements of H by (u;a) where u € H~ and a E H.

CLAIM (1): 3. l:ii":ii!l+l V !l (S(U;o))(k) = o • J' J u€H

Suppose this were not t~ue, viz.

(2.6) V. l<'q 1 3 !l: (S(VJ. ;o))(j) r o • J, -J- + v.EH

J

Il Then there exists u1 € H such that (S(u1;o))(1) po. We proceed by

induction. So, let us assume that there exists uk E H!l, such that

(S(Uk;o))(j) p o, 1 :il j :il k.

If (S(uk;o))(k+1) p o, then we take uk+l = uk.


If not, then by (2.6) there exists vk+1 € W such that (S(vk+t;o)} (k+1) F o.

Now take ~+t > 0 so small that for 1 ::ii j :ii k

We set uk+1 •uk+ Àk+lvk+l* Then for 1 :ii j :ii k+1. (S(Uk+l;o))(j) >/= o.

So our assumption finally leads to UR.+l E H1 with the property

(S(uR.+l ;o))(j) F o, 1 :ii j :ii R.+1.

However, we obtain a contradiction:

and this proves Claim (1). So there exists j0

, 1 :ii j0

:ii R.+1, such that

v 1

: (S(u;o))(j0) • o • u€H

Let tjl 1 denote the permutation of { 1 , ••• , t+ 1}, which only interchanges

j 0 and R.+1. Then. clearly,

V 3 : (~1 c S)(u;o) u€Ht ÜEHR.

(Ü;o) •

CLAIM (2): vaEH 3aEH : (~1 0 S) (o;a) = (o;a) •

To prove this claim, let a E H. Put (w;a) ... (Ïf;1

o S)(o;a).

Take u0

E HR. such that u0(j) i: o, 1 :ii j :ii R.. Put (u

0;o) = <1P

1 o S)(u0 ;o).

For p > 0 let ap = ~R.+t((u0 ;pa),(u0 ;pa)). Since <$1 o S)(u0 ;pa) = <'U'o+pw;pa), we get that

Hence w • o. This proves Claim (2). We have also derived

So there exists rR.+l > 0 such that


vaEH 3aEH: <$1 • S)(o;a) "'(o;i') and llallH"' rQ.+tftallH.

We define s1

E GR, and the unitary operator RR,+l on H by

+1/Q.rv -1 ,.., s1 u = ri+l u and RQ.+l a = rQ.+l a

where ($1

o S) (u;a) "" ""' Q, (u;a1, u € H-, a E H. Hence

The induction hypothesis implies that there exist a permutation iJ! of

{1, ••• ,t}, unitary operators Rj, 1 :;; j :;; Q., and r0

€ IRQ, with Q,

nj=l r0 (j) = 1 such that s1 = îji o [R1© ••• 0RQ.] o Dr0

•

So,

(ij'j1

• S)(u;a)

We consider ij; as a permutation of {1" •• ,Q.+1 }. Let q:> = ip 1 ° iJ! and let

r € IR.R.+l be defined by r(j) "' r0 (j)r;1~Q,' 1 :;; j :;; t, and r(Q.+1) rQ.+l"

h R,+ 1 (.) Note t at Tij=l r J = 1.

Finally, we have S = <'P 0 [R10 ". 0RJ(,+l] • Dr. o

Next we introduce the n-fold symmetrie Hilbert tensor product of H. To

this end we introduce some linear operators in Tn(H). Let En be the n-th

order permutation group. For each permutation a € En the unitary operator

a on T (H) is defined by a(F) = F o 6, F E T (H). n n

The n-folil symmetrizer S(n) is the orthogonal projection on T0

(H) defined

by

S(n) = ~! l aEE

n

The range of S(n) is called the n-foZd syrrmetric tensor product of H and

is denoted by T5 ym(H), i.e. n

T8 ym(H) "'S(n)(T (H)) • n n


Being a closed subspace of T (H), the space Tsym(H) inherits the Hilbert n n space structure of T (H). We rem.ark that

n

Tsym(H) = {F € T (H) 1 V €" : Ü(F) = F} • n n a "'n

Further, put T~ym(H) • t. The Hilbert spac.e T:ym(H) is itself a func

tional Hilbert space:

2. 3. Lemma

Let n € lN. Let [Q9 n] E PT(ff) be defined by sym

©n [Q Jsym (x,y) x,y € ff .

Then

Proof:

Apply Lemma 1.12.

The elements in Tsym(H) are completely determined by their restrictions n

to the diagonal of Hn.

2.4. Lemma

For all n € lN we have

V : (VxEH : F(x, ••• ,x) = O) • F = 0 • FET8ym(H)

n

Proof:

The proof is by induction. For n = 1 the assertion is trivially true.

Suppose that the assertion is true for n = R..

Let F E T~~(H) such that VxEH: F(x, •••• x) = O.

Let a,b € H. For all À E 4: we Pa.ve

R.~ 1 (R.+1) R.+1-k 0 = F(a+Àb, ••• ,a+Àb) L k À F(a, ••• ,a, b, ••• ,b) k=O k R.+1-k

Hence, in particular, F(a, ••• ,a,b) = O.

We define the R.-linear function Fb by

c

50 II .Functional Hilbert spaces

Fb(x) • F(x(1}, ••• ,x(i),b) , i x E H •

Since Fb belongs to T~ym(H), our induction hypothesis implies Fb • 0.

Hence F • 0,

Next we introduce the symmetrie Fock space F (H). sym

(2.7) The symmetrie Fock apace F (H) of H is defined as the direct sym sum

00

F (H} • ID Tsym(H) • sym n•O n

The elements in F (H) will be denoted by F or (F(n)} , where sym - - n F(n) E T8ym(H), etc. For example: - n

00

Our next goal is a functional Hilbert space representation on H of

F (H) • sym

2. s. Definition

For each x E H we define € F (H) by sym

For eacb F E F (H) the function W(!_) on H is defined by sym

For all F E F (H) and x E H we have - sym

(W (.!)) (x)

In particular

(W(e } )(x) __,,,

00

l ~1~ (F(n)}(x, ••• ,x) n-0 rnr -

0


2.6. Lemma

The linear mapping W from F (H) into the space of complex valued funcsym

tions on H is an injection.

Proof:

Let !, € F (H) with W(F) = O. For all x E H and À E ~ sym -

oo Àn (W(!_))(Àx) = L (!(n))(x, •• ,,x) = 0 •

n=O liîf

Hence, for alle n ~ O, Vx€H : (!_(n))(x, ••• ,x) = O.

Because of Lemma 2.4: !_(n) = o.

2. 7. Definltion

The Hilbert space S(H) is defined to be the space W(F (H)) with inner sym product

So S(H) is a representation of F (H). sym

2. 8. Theorem

S(H) = H(H, exp[Q]) •

Proof:

For all F € S(H) and y € H we have

F ,G € F (H) • -- sym

So S(H) is a functional Hilbert space with reproducing kernel

<x,y) 'ti> (W(!y)) (x) = exp(x,y)H •

2. 9. Definition

For n ~ 0 the Hilbert space S (H) is defined to be the closed subspace n

W(T:ym(H)) in S(H) with induced inner product.

a

a


2.10. Theorem

00

a. S(H) 9 s (H) • n=O n

b. S (H) = H(H _!_ Qn) n • n!

Proof:

a. The operator W is a unitary operator from F (H) onto S(H). sym

b. For all FES '(H) and y € H we have n

The reproducing kernel of Sn(H) is

(x,y) >+ (W(y©n//iiÎ)) (x) 1 = n!

From now on we often identify S (H) and T6 ym(H), S(H) and F (H). In n n sym view of the previous theorem this agrees with the definition of the

symmetrie Fock space in (2.7).

We state some results on the elements of S(H).

2. t 1. Lemma

0

Let F E S(H) and let, for all n ~ O, Fn denote its orthogonal projection

on S (H). n

Then F(x) = I:=o Fn(x), x € H, where the series converges absolutely and

uniformly in x on bounded sets in H.

Proof:

For all n ~ 0 and x € H :we have

For m € JN we have

00

l D

n=m

Il Functional Hilbert spaces

2.12. Corollary

2.13. Lemma

Let F E S(H).

The function F is analytic on H.

Proof:

Let F " -r"'=O F with F E S (H). Ln~ n n n

Fix n > O. Then Fn (x) = (F, ~! (Qn)x)S(H)'

Since for all a,b € H and À € ~

x E H.

(where all terms in the sum belong to S (H),) n

53

the function Fn is ray-analytic as well as continuous on H. Hence Fn is

analytic on H. Since the series F = I:'.0 Fn converges absolutely and

uniformly on bounded sets, the function F is analytic on H. a

Finally we give an orthonormal basis in S(H). We employ the multi-index

set lM(D>) as introduced in Definition I.3.2.

2. 14. Definition

For each s E lM(D>) we define the function ~s on H by

1 s (') ~ (x) =- n (x,e.) J s rsT j€D>[s] .J H

x E H •

Let s € lM(D>) with Is 11 = n. Then t:here exists (ek " •• ,ek ) E ~ such 1 n

that

(nl)t '4>5 = Sï W[S(n)[ek © ".0ek ]]

1 n

54 II · Functional Hilbert spaces

2.15. Theorem

a. For n ~ 0 the set {~5 I s € lMn(ID)} constitutes an orthonormal basis

in s (H). n

b. The set {~5 1 s € JM(lD)} constitutes an orthonormal basis in S(H).

Proof:

a. Let n ~ 0 and let s,t € lM (ID). Let n

and

Then

a = <~s.~t)S(H) =

= li!(s!t!f6 (Cek 0."©ek ],S(n) [e.Q, 0 ••• ©e.Q, l}T (H) =

1 n 1 n n

Let s 1 t, say s(j) > t(j). Then at least one e. = ek· has to be J JO

paired with e. = e.Q, 1 e., so all terms in the sum vanish. Hence

0 i. o(jo) J

Cl. = • Let s = t. Then we have

a. = (s!)-1

# fo €En 1 VjEID: s(j) = #{m 1 1 ::iîm:>n, j= R.o(m)}} =

(sl)-1 si = 1 •

Hence {~5 1 s € lMn(ID)} is an orthonormal system in Sn(H). Since for

all x,y € H

r DYY r 1 s-s n l s Y ~s(x) = l Sïx (y) "'n! (x,y)H'

SEJM (ID) SEJM (ID) n n


Lemma t .11.c. implies statement a.

b. Statement a. and Theorem 2.10 yield b.

From Theorem 3.28 it follows that there exist orthonormal bases of

harmonie homogeneous polynomials in S(H).

§ 3. Examples of functional Hilbert spaces

This section deals with two classes of functional Hilbert spaces:

1. The Bargmann spaces, consisting of analytic functions on Hilbert

spaces.

2. Functional Hilbert spaces, consisting of harmonie functions on IRq,

q € JN.

§ 3.1. The Bargmann spaces

We consider Bargmann spaces both on finite and infinite dimensional

complex Hilbert spaces.

The usual norm and inner product in (:q, q € JN, will be denoted by

1·1 2 and (•,•) 2, respectively.

3. 1. Definition

Let q € JN.

The Bargmann space of order q, denoted by B , is the functional Hilbert q

space, consisting of all analytic functions F on (:q, for which

J (FCz>! 2 exp (- lzli)dxdy <co,

(:q

The inner product in B is given by q

• 'IT-q J F(Z)G(Z) exp (- lz li) dX dy

(:q

(Z = X + iy, x,y € IRq • )

D


A detailed treatment of these spaces can be found in Bargmann 1 s paper

[Ba 1].

The function K in PT(éq) is defined by K exp[(•,•) 2]. We recall q q

that H{(q,K ) = S(éq). q

3.2. Theorem

Let q € lN.

Then B = S (Cq). q

Proof:

The functions ~s: (z 1+ z5/1Sï, z € C:q), s € lN~, constitute an ortho

nonnal basis in B. Cf. [Ba 1], (1.6). For all z,w € C:q we have q

Lemma 1.11.c. implies that B = H(Cq,K ). q q

Let q € 1N and n ~ O. Set

1 n q Kq,n = n! (•,•) 2 E PT(C) and

The spaces B , n ~ 0, consist of all n-homogeneous polynomials on tq. q,n They are closed and mutually orthogonal in B and they satisfy

00 q Bq = CDn=O 8q,n·

The dimension of B can be calculated with the aid of its reproducing q,n kemel. Indeed:

3. 3. Lemma

Let q € 1N and n ~ O. Then

dim(B ) = (n+q-1) q,n q-1

Proof:

Let d = dim(B ) and let {~. 1 1 ~ j < d+1} denote an orthonormal q,n J 2

basis of B • Let s2 1 denote the unit sphere in IR q and let dcr2 1 q,n q- q-denote its (2q-1)-dimensional surface measure.

0


For all Z E t we have

Hence

d 2 1 2 Ï llJ!.(z)I • 1 lzl n

j=l J n 2

= 1T -q ~ 1 J 1 dcr2q-l

s 2q-1

00

J r2n+2q-1

0

-1 1 2'1fq +q 1 = 1T -, F" !r(n+q) = (n - ) n. l \QI . q-1

2 exp(-r )dr =

We present a result on the inner product in B • q

Let Pol(Cq) denote the space of all polynomials on Cq. For each

F E Pol(Cq) we define the polynomial F by F(z) = F(ZIT), ••• ,Z(q)) a a and the differential operator F (ó) by F (1iïîIT , ... , "äzTciî).

3. 4. Lemma

Let F,G. E Pol{Cq).

Then (F,G)B = [F(3)G](o). q

Proof:

For S E JN!, let IJ.15: Z 1+ z5 , Z E éq. For all s, t E JN! we have

Since Pol(éq) = <{~5 1 SE JN~}>, the statement follows.

The BargJ!lann space B can also be described as a functional Hilbert q

space on IRq. Consider IR.q as a subset of Cq.

57

0

0

58

3.5. Lemma

Let q E lN.


The mapping F 1+ Fl:mq• F E Bq' is a unitary operator from Bq onto

H(lRq,Lq).

Proof:

From Lemma 1.14 it follows that we only have to prove that the mapping

F 1+ Fi:mq• F E Bq, is an inject~on. Since the elements of Bq are ana-

lytic functions, the mapping F -+ F!:mq• F E Bq' is an injection. c

We mention some consequences.

(3.1) The mapping F 1+ F!:mq• F E Bq,n' is a unitary operator from

B on to H(Ill ,L ) • q,n q q,n

00

(3.2) H(Illq,L ) = @ H(Illq,L ) q n=O q,n

(3. 3) The set {x 1+ Xs //Sï, x € :mq j S € lN1} is an orthonormal basis

in H(Illq,Lq).

(3.4) Let again Pol(Illq) denote the space of all polynomials vn :mq.

For each F E Pol(IRq) we define the polynomial F by F(x) = FlxJ and the

differential operator F(D) by F(D) = F(~ , ••• , ax~n))' As in Lemma 3.4 we have for F,G € Pol(IRq)

The Bargmann space of infinite order is the last subject in this sub

section. We introduce some notations and conventions.

We write CJ1.2 instead of ~2 (lN). The inner product and norm in t~2 are

denoted by (o , -) 2 and ! • 12, respec tive ly.

The set {ej 1 j E lN} is the standard orthonormal basis in tJ1.2•

Let IRJ1.2 be the set {z E U 2 1 VnElN: z(n) E IR}.

Identification of f:q and the linear span <{e. 1 1 :;ii j ::ii q}> yidds the J

scheme:

II Functional Bilbert spaces

lRq c: lRq+l

n n cq c: 4:q+1

For all q € 1N we have found Bq • S (f:q). So, the next definition is

obvious.

3. 6. Deflnltion

The function K.., in PT(U2

) 'is defined by K00

• exp[(•,•) 2 ].

We define the Bargmann space B00

of infinite order as the space

S(tt2) • H(tt2,K00).

Let us consider 800

, introduced in [Ba 3), in some detail.

(3.5) For n ~ 0, let 800

= H{U2,~ C-,•)2n) = S (t.t2). ,n n. n

The spaces 800

n are closed and mutually orthogonal subspaces of B00

•

' 00 They satisfy 800 = C9 _n B • n-v oo,n

3. 7. Deflnition

Let q € JN.

Let T denote the orthogonal projection from t.t2 onto tq. q .....

The function Kq € PT(tt2) is defined by

For F € 800

the function Zq(F) on t.t2 is defined by

(Z {F))(Z) = F(T Z) , q q

3. 8. Lemma

Let q € lN.

59

The operator Z is an orthogonal projection on B with range H(f:t2,'K ). q co q

Proof:

Let F € 800

• For R. € 1N and a.. € C:, W. E C:t2, 1 :il j :il .t, we see that J J

R,

1 I a..(Z (F))(w.)1 2 j•1 J q J


Since Corollary 1.21 implies K ~ K, we get tbat Z (F) EB and q 00 q 00 2

llZq(F)ll500

~ llFlls00

• So llZqllB00

::> 1. Further it is obvious tbat Zq = Zq.

These two observations yield tbat Z is an orthogonal projection. Cf. q [Yos], Theorem III.3.3.

By LenD!la 1.12 we have Zq(B00

) = H(C22,Kq). o

We denote Zq(B:e) by sq. The mapping F I+ Fltq' F E Bq, is a unitary

operator f rom B onto B • q q

3. 9. Lemma

Let F E 8 . 00

Then lim Z (F) "' F • q..- q

Proof:

The bounded sequence (Zq(F))qElN bas a weakly convergent sequence with

limit F. From this the statement follows illllllediately. o

3. 10. Corollary

Let F,G E 800

• Then

(F,G)B = lim n-q 00 q-+oo

Proof:

For q E lN .we have

J F(z)"'G(i) exp (- lz 1 ~) dx dy •

tq

(Z (F)~Z (G))B = n-q q q 00

/F(Z)G(Z) exp (- lzl~) dxdy •

tq

Cf. [Ba 3), (9).

3. 11 . Corollary

Let F denote a continuous function on' 4:22 such that

V ElN : F 1 t is analytic on tq , q q

a = sup (n-q /IF<z>i 2 exp(- lzli)dxdy) < oo

qElN tq

0

II Functional H.ilbert spaces 61

2 Tben F € B.., and llFlls.., = a..

Proof:

For q € lN there exists F € B such that F = F o T • The sequence q q q q (F ) €lN is bounded in B and bas a weakly convergent sequenee with q q 00

limit G € 800• It is o~ious that F = G € 800• Since a. = :~~ (llZq (F) Il~), we obtain that a. = llFll&x.. c

We derive the infinite dimensional analogue of Lemma 3.5.

The funetion L.., in PT(IRR.2) is defined by L00 = Keel IRR. x IRR. • 2 2

3.12. Lemma

The mapping F i+ FIIRt , F € B, is a unitary operator from Beo onto 2

H(IR.11,2

,L00).

Proof:

From Lemma 1.14 it follows that we only have to prove that the mapping

F i+ Flm.11,2

is injeetive. Let F € Beo with FIIR..11,z = O. Sinee for eaeh

q € IN the restrietion FIIRq = O, it follows that Zq(F) = O. Hence

F = lim Z (F) = O. c q-KIO q

§ 3. 2. Hilbert spaces of harmonie functions

For q ~ 2 we introduce a Hilbert space of harmonie functions on IRq,

which will turn out to be a subspace of H(IRq,Lq).

3. 13. Definition

.Let q ~ 2.

By H we denote the spaee of all harmonie functions F on IRq which q

satisfy

/1F<x>l 2 exp <- llxli)dx < 00•

IRq

The inner product in Hq is defined by

J F(x)"'G(XJ exp (- j lx li)dx •

IRq


In the sequel we use the following notations:

S , the unit sphere in IRq, q-1

doq-l' the (q~1)-dimensional surface measure on Sq-l'

a " q-1

. We state the Mean-Value Theorem for harmonie functions.

3.1 IJ. Theorem

If F is an harmonie function on IRq, y E IRq and r > O, then

F(y) = a1- / q-1 s

q-1

F (y + rl;)do 1 (~) •

q-

Next we show that H is a subspace of H(IRq,L ). q q

3.15. Lemma

Let F E H • q

a. V F(a) aEIRq

(2rr)-iq JF(x) exp (- !lx-al;)dx •

IRq

b. The function F on Cq defined by

F(a + ib) exp{!lbl; + i(a,b) 2)

(27f) lq J F(X) exp (- !lx-a1; + i(x,b) 2)dx ,

IRq

q q ,.. a ,b E IR , is analytfc on C and F 1 IRq "' F.

Proof:

(2ir)-iq JF(x) exp (- llx-al~)dx =

IRq


= (2n)-!q /F<a + u) exp (- ! lu l~)du lRq

) 2 q-1 F(a + r!:;)doq-l (!:;) exp (- ir )r dr = F(a)

because of the Mean-Value Theorem.

b. We only remark that for z E ~q

63

F(z) = (2n)-iq /F(X) exp (- i lxl~ + (z,x) 2 - ! Ï z2(j)) dx • a j=1

lRq

3. 16. Theorem

Let q ;;: 2. Then

a. The space H is a functional Hilbert space. q

b. H c H(lR\L ). q q

c. VFEH : llFllH(lRq L ) q ' q

Proof:

Let F € H and let F be the analytic extension to tq, defined in Lemma q ~

3.15.b. We show that F belongs to B. Then q

n-q /IF<z>l 2 exp (- lzl~) dxdy

tq

= n-q /( J 1(2'1T)-lq J F(x) exp(-llx-al~ + i(x,b) 2)dxj2

db)·

lRq lRq lRq

• exp (- la l~)da •

The Fourier transfonn is a unitary operator on L2(lRq), so the latter

integral equals


n -q J j 1 F ( x) 12 exp (- 1 x-a 1; - 1a1 ;) dx da

lRq ]Rq

TI-q J /!F(x)l2

exp(-ilxl;- 21a-!xl~)dadx lRq lRq

= TI-q J IF(X) 12 exp (- ! lxl;) (!n) !q dx = llFll~q •

lRq

Lemma 3.5 implies that F E H(lRq,Lq) and that flFllH(lRq,Lq)

So statements b. and c. are proved.

From c. it follows that for all F E Hq and y E lRq

2 ~ llFllH • exp (- ! ly 12) •

q

So H is a closed subspace in H(lRq,L }, whenee statement a. follows. q q Let F E H(lRq,L ) be a harmonie function and let F denote its projec-

q n tion on H(lRq,L ). Since F(ÀX) \

00

0 Àn F (x), it easily follows q,n l.n= n that the F 's are harmonie, n-homogeneous polynomials. Henee F E H

n q an,2: statement d. is proved. o

The results we present in the ramaining part of this subseetion, start

in [Mu], [Vi], [MOS].

3. 17. Definition

Let q ~ 2 and n ~ O.

The space H is defined to be the spaee H n H(lRq,L ) endowed with q,n q q,n the induced inner product.

REMARKS:

(3.6) The spaee H is the space of all harmonie n-homogeneous poly-q,n nomials on lRq.

6) H n=O q,n

(3.8) For harmonie polyn6mials F and G on IRq we have

(F,G)H = [F(D)G](o) q


3.18. Lemma

Let q <;; 2, n ;:: 0 and write d = dim(H ) • q,n q,n Tben we have

d = (2n+q-2)f(n+q-2)/r(n+1)f(q-1)) q,n , n <;; 1 , and d 0

= 1 • q,

Proof:

See [Mu], pp. 3 and 4.

3.19. Lemma

Let q ~ 2 and let n ~ O. Let F,G € H • Then q,n

Proof:

We have

3. 20. Definltion

FCO'G'<U do 1 (t) • q-

f F(X)G(X) exp (- llxl~)dx =

IRq

"° = (21r)-iq f f

0 s 1 q-

2° f (n+iq) f " 21flq s q-1

n

J 2 f(n+lg)

" r(Jq) ()' 1 q- s q-1

F (t)G(t) do 1 (t) r 2n+q-l e -h2

dr = q-

F(t)G(t) dcrq_1 (t)

F(t)G"(U do 1 (t) • q-

Let q <;; 2, n ;:: 0.

M and M are the reproducing kemels of H and H • q q,n q q,n

We want an explicit expression of these reproducing kemels. The case

q = 2 is easy to handle, because d2 = 2, n <;; 1. Only the results are 2 ,n

mentioned here. For x,y € IR we have

0

0


M2 0

cx,y) = 1 , '

M2 n (x,y) = 2 Ref-1- (Cx,y) 2 + i det(x,y)) n] , n ;;: 1 , ' 2n nl

M2(x,y) = 2 exp CHx,y) 2) cosO det(x,y)) - 1 •

First we determine the reproducing kemels M , q ;;: 3, n ~ 0. q,n À

To this end, we introduce the Gegenbauer polyn0111ials C of degree m, À m

m ~ O, and order À, À > O. The polynomials Cm are generated by the

function

(3.9) 2 -À

(1-2tr+r) I t E [-1, 1] , lr I < 1 • n=O

From relation (3.9) it follows

(3.10) CÀ(1) = f(n+2Ä)/(nl f(2Ä)) • n

The explicit expression for the Gegenbauers is

(3.11) CÀ(t) 1 [n~2] (-1)m f(n-m+Ä) (2t)n-2m • n = f(Ä) m;O ml (n-2m) !

We also mention the 'inverse' formula:

tn = ~ [n~Z] (n-2m+À) CÀ 2

(t) (l.lZ)

2n m~O m! f(n+1-m+À) n- m

3.21. Lemma

Let q ~ 3 and n ~ O. Let {H. I 1 ~ j ~ d } be any orthonormal basis J q,n

in H • Then for ~.n € S 1 q,n q-

d

tn H.{ri)H.(~) = f0q-1) c!q-1 (C~,O)z) • j=1 J J 2n r(n+!q-1)

Proof:

{(2n f(n+iq))l I < . } By Lemma 3.19, the set r(lq) Hj 1 = J ~ dq,n is an ortho-

normal basis with respect_to the inner product


(F ,G) ~ - 1- f F(~)G\U" da 1 (~) •

(jq-1 q-s q-t

By [Mu], Theorem 2 and Lemma 18, we find that for all ~.n € Sq_1

Lemma 3.18 yields

d n ~ H:ëTiYH.(~) = r(iq-1) c!q-1 (<~.n> ) . j•1 J J 2n r(n+jq-1) n 2

3. 22. Theorem

Let q ~ 3 and let n ~ O. Then

a. For all x,y E lRq, x # O # y,

M (x,y) " r(lq-1) ciq-1(( x , Y ) .) lxln IYln • q,n 2n r(n+!q-1) n ixr; 1YT; 2 2 2

b. For all x,y E lRq

M (X,y) q,n

67

[n/2] (-1)m r(n-m+lq-1) 2m 2m 2m "----- mt ml (n-2m) ! (2(x,y)2)n- lxlz IYlz •

2n r(n+iq-1) -v

Proof:

a. This statement is a corollary of Lemma 3.21.

b. Use relation (3.11).

3. 23. Corollary

Let q ~ 3 and n ~ 0. Then for a];l x,y E lRq

n1'· (x,y)2n • [nf2] r(n-2m+lq+1) M (x,y)lxl2mlyl2m •

m=O 22m r(n-m+iq+1) q,n-2m 2 2

Proof:

Use relation (3.12).

c

IJ

IJ


Let the m-th term in Corollary 3.23 be denoted by N (x,y). As sets 2 2 q,n,m

we have H(lRq,N ) = {x,... lxl2m G(x), x E lR 1 G E H 2m}. With the q,n,m q,n-

aid of property (3.8) it can be shown that

[n/2] H(lR2 ,L ) = 19 H(lR\N )

q,n m=O q,n,m

This relation expresses the well-known fact that each n-homogeneous

polynomial Pn can be written as

Pn(x) 2 2[!n]

~(x) + lxl Qn_2(x) + ••• + lxl ~-2 [!n](x)

where Q 2 ., 0 ~ j ~ [!n], denotes a harmonie, (n-2j)-homogeneous polyn- J

nomial. Cf. [Vi], Chapter IX, Section 4.

Let O(q) denote the orthogonal group on lRq and let IT be the repre-q,n -1 .

sentation of O(q) in H(lRq,L ) given by II (R)F = F o R , R € O(q), q,n q,n F € H(lRq,L ). It is clear that the spaces H(lRq,N ) remain in-q,n q,n,m variant under the unitary operators R»- Il (R), R € O(q). q,n

The representations Rt+ IT (R)IH(lRq N )' R € O(q), are irreducible. q,n ' q n m

Cf. [Vi], Chapter IX. ' '

As a corollary to Theorem 3.22 we mention

3. 24. Corollary

Let q 3 and n ;;; O. Let F E H For all x E lRq q,n

IF<x>l2 llFll~ rog-1 )f(n+g-2)

lxl 2n ~ r(n+iq-1)r(q-2) 2n ' q n.

Proof:

For all x E lRq we have IF(x)l 2 ~ llFll~ M (X,x). The result.follows q q,n

from Theorem 3.22 and relation (3.10) with À= iq-1. a

The next theorem presents an explicit expression for the reproducing

kemel M of the functional Hilbert space H . q q


3. 25. Theorem

Let q ~ 3. For all x,y E IRq we have

x"' 0 'Î' y

Mq(x,y) = 1 , x = o or y = o •

Proof:

. "" ~"" Since H = ti 0

H , Lemma 1.11.c. implies that M = M q n= q,n q =O q,n •

Let F E Hq. Since F E H(IRq ,Lq), we have j F(X} j i llFllH exp(!jx I~), x € IRq. We give another uniform estimate on m.q. q

3. 26. Corollary

Let q ~ 3.

For all p > 1 there exists y > 0 such that q,p

Proof:

Let F E Hq and let X E IRq. Bij relation (3.1.0) we get that

00 lx1 2n l jF(x} 1 s llFll (1 r(!q-1}r(n+q-2) _2_) •

- H =O r(n+lq-1)r(q-2) 2n 1 q n.

r(!q-1)r(n+q-2) < 2 pn Let p > 1. There exists Yq,p > 0 such that r(n+tq-l)r(q-2) Yq,p •

69

c

Hence the assertion follows. c

Finally, we consider harmonie functions in 800

, the Bargmann space of

infinite order. We shall prove that U H is dense in 800

• We recall

that the operators T and Z have bee~lnt;oduced in Definition 3.7. q q

3. 27. Definltion

A function F € 800

is called a harmonia funation if there exists q E JN

such that F = Zq(F) and the restriction FiIRq is harmonie on IRq.


3. 28. Theorem

The space of all harmonie functions in 800

is dense in 800

•

Proof:

For q ~ 2, let

If U V is dense in H(lRJl.2 ,L ). then Lemma 3.12 implies the statement. q2:2 q co

With reproducing kernel theory we prove that indeed U V is dense in q~2 q

H(lRt2,L,).

Let M be the reproducing kemel of v . Since vq c vq+1 c H(lRt2•L,x,), q ,..., ~ q

we find that Mq ::> Mq+l ::> L00 • Because F 1+ Fimq• FE Vq' is a unitary

operator from V onto H , we obtain that q q

00

Mq(x,y) = M (T x,T y) l M (T x,T y) q q q n=O q,n q q

Fix x,y E lRt2 and n ;s; 0. Corollary 3.23 impliès that

M (T x, T Y) :;; -1, IX 1 n2

• IY I n2 q,n q q n.

and Theorem 3.22.b. implies that

1 n lim M (T x, T y) = n! (X,y) 2 q-><x> q,n q q

Hence

By Theorem 1.22 the union U V is dense in H(lRt2,L00).

q§!2 q

In [Ma 11 the following results can be found. They are based on the

previous theorem.

(3.13) A generalization of the Mean-Value Theorem for harmonie func

tions: For all F € H(lRJl.2,L"',), X E 1Rl2

and r > 0

F(X) = lim - 1- /· F(T X + r~) do 1 (~) •

q-><x> oq-1 q q-S q-1

(Cf. [Ma 1], Theorem 3.22.)

tJ


(3.14) A Weak version of the min-max principle of harmonie functions:

Let D be a bounded open subset in lRR-2 with boundary r such that D is

weakly closed and let F E H(lRt2 ,L~). Then

VXED IF(x)I ~ sup IF(y)I • YEf

Cf. [Ma 1], Theorem 3.15.

(3.15) A construction of an orthonormal basis of harmonie polynomials

in B~. Cf. [Ma 1], Definition 3. 17.

Appendix

In this appendix we consider some linear operators in the symmetrie

Fock space S(H) of a Hilbert space H. The operators we study, are the

creation and annihilation operators, dilatation operators and second

quantized operators. Reproducing kernel theory plays an important role

in this appendix. We feel inspired by [Ba 1].

Recall that Sn(H) = H(H, ~! (•,•)~), n ~ O, and S(H) = H(H,K), where

71

K = exp[(•,o)H]. As usual, V(H,K) denotes the linear span <{Ky 1 y € H}>.

We start with the annihilation and creation operators•

A. 1. Deflnltion

Let u € H.

The annihilation operatoP a(u) in S(H) is defined by

[a(u)F](x) = lim (F(x+Àu) - F(x))/À, À-t-0

where F is in the maximal domain D(a(u)).

The aPeation opeI'atOP y(u) in S(H) is defined by

[y(u)F](x) = (x,u)H F(x) , x E H ,

where F is in the maximal domain D (y(u)).

x € H ,


REMARKS:

(A.1) For all u E H the space V(H,K) is contained in D(a(u)) as well

as in D(y(u)),

(A.2) Let u E H. The annihilation operator a(u) maps Sn(H) into

S 1

(H) (o:(n) annihilates a particle) and the creation operator y(u) n-map s S (H) into S

1(H) (y(u) creates a particle). With reproducing

n n+ kemel theory it can be shown that

\>'FES (H) n

(A.3) For all u,v E H we have the Canonical Commutation Relation

a(u)y(v) - y(v)a(u) = (u,v)H I ,

where I denotes the identity on S(H).

(A.4) For all u,x € H we have

(a(u)F) (x)

(y(u)F) (x)

A.2. Theorem

Let u € H.

(F • y(u)Kx) S (H)

(F ,o:(u)Kx) S (H)

F E D(a(u))

F € D(y(u))

The operators o:(u) and y(u) are mutually adjoint.

Proof:

We only give a sketch of the proof.

By (A.4) we derive D(y*(u)) cD(a(u)) and D(a*(u)) c D(y(u)).

So we only have to prove

(A.S) VFED(a(u)) VGED(y(u)) : (a(u)F,G)S(H) = (F,y(u)G)S(ff) •

We use a smoothness argument. Let for À, 0 <À< 1, the bounded operator

ZÀ: S(ff) + S(H) be define~ by (ZÀ F)(x) = F(Àx), x € H, F E S(ff).

Remark that lim ZÀ(F) =F. Then we find: Àt1


(A.6) For, small À the operator a(u) o ZÀ is bounded in S (H).

(A.7) For F E D(a(u)) we have

- ZÀ F € D(a(u)) ,

- (a(u) o ZÀ)F ""a(u)F, À t 1, weakly,

For all GE S(H) the function À tt- (<a(u) 0 ZÀ)F,G)S(H) is extendible

to an analytic function on the unit disc.

For À, 0 <À< 1, HE V(H,K) and GE D(y(u)), we find that

(ZÀ H, y(u)G)S(H) = (fo(u) 0 ZÀ)H,G)S(H) •

(A.6) implies that for small À, for F E D(a(u)) and G € D(y(u))

Application of (A. 7) yields statement (A.5).

73

a

Using relation (A.3) and a similar smoothness argument it can be proved

that for all u E H D(a(u)) = D(y(u)).

We consider the exponentials of a(u) and y(u). For u,v E H we have the

relations

so

(exp(a(u))Kv)(x) = (exp((u,v)H)Kv)(x) = Kv(u+ x) ,

(exp(y(u))Kv)(x) = exp(x,u)H Kv(x) ,

exp(a(u)) = Tu and exp(y(u)) = Ru

This leads to:

A. 3. Definition

Let u € ff.

The exponentials of a(u) and y(u) are as operators in S(H) defined by

(exp(o.(u))F)(x) = F(u+x) , x E H,

where Fis in the maximal domain D(exp(a(u))),

(exp(y(u))G)(x) = exp[(x,u)] G(x) , x € H,

74 II Functional H:Hbert spaces

where G is in the maxim.al domain D(exp(y(u))).

(A.8) For all u E H

(exp(a(u))F) (x)

(exp(y(u))F) (x)

A. 4. Theorem

Let u € H.

(F,exp(y(u))Kx)S(H)

(F,exp(a(u))Kx)S(H)

F E D(exp(a(u)))

F E D(exp(y(u)))

The operators exp(a(u)) and exp(y(u)) are mutually adjoint.

Proof:

Cf. the proof of Theorem A.2.

The second type of operators we study, are the dilatations.

A. 5. Definition

Let A E B(H).

* The ditatation Z(A) in S(H) is defined by (Z(A)F)(x) = F(A x), x E H,

where Fis in the maxima! domain D(Z(A)).

(A.9) For all A,B E B(H) and all y E H we have

- Z(AB) = Z(A)Z(B),

- Z(A)Ky == K(Ay) ,

- The operator Z(A) maps S (H) into S (H) and the operator z,(A) re-n n stricted to Sn (H) has norm llAll~,

- If llAllfi ;;;; .1, then llZ(A) llS(H) ;;;; 1.

A. 6. Theorem

Let A E B(H).

Then (Z(A)) * = Z(A*) •

Proof:


Finally we consider secon~ quantized operators. The second quantization

operator Q is a mapping, which associates to an operator A in H an

0

0


operator Q(A) in S(H) such that (Q(A))* = Q(A*). We call the operators

Q(A) second quantized operators.

A. 7. Definltion

Let A E B(H).

The seaond quantized operator Q(A) in S(H) is defined by

(Q(A)F)(x) = (F,y(Ax)Kx)S(H) , x E H ,

where F is in the maximal domain D(Q(A)).

(A.10) For all A E B(H) and y E H we have

* Q(A)K = y(A y)K • y y

A. 8. Theorem

Let A E B(H).

Then (Q(A))* = Q(A*).

Proof:


A. 9. Theorem

Let A be a positive self-adjoint operator in B(H).

Then Q(A) is a positive self-adjoint operator in S(H) and etn(A) = Z(etA), t E lR.

Proof:

With a smoothness argument it can be proved that Q(A) is a positive

self-adjoint operator. By Theorem A.6 the operator Z(etA) is self

adjoint.

We only have to prove that etQ(A) c: Z(etA), Let F E S(H), x E H, and

let t > 0. < ( -Tril(A) ) For 0 ;:> T - t put f(T) == e F, K(eTAx) S(H)' Then, because of

relation (A.10), for 0 < T < t

( -T:i'2(A) ) f' (î) = - S'l(A)e F, K(eîAx) S(H) +

( -î'2(A) îA ) + e F, y(e Ax)K(eîAx) S(fl) = O.

75

0


Hence f(O) = f(t), whence for all x € H

So

h . h . 1 . -tQ(A) Z( -tA) w ic 1mp 1es e = e • It easily follows that etQ(A) = Z(etA). 0

lntroduction

CHAPTER 111

INDUCT IVE AND PROJECTIVE LIMITS

OF SEMI-INNER PRODUCT SPACES

As stated in its title, the main goal of this monograph is the intro

duction of spaces of analytic functions on sequence spaces. To be more

specific, we consider inductive limits and projective limits, Find[g]

and F .[g], of (semi)-inner product spaces consisting of analytic pr~ +

functions on i.p(ID). Here g is a subset of w (ID) of a special type with

w•(ID) the set of all positive sequences on ID.

The introduction of F. d[g] and F .[g] involves sequences spaces over in prOJ

ID and sequence spaces over JM(ID). (Recall that JM(ID) denotes the set

of multi-indices over ID.) All mentioned sequence spaces admit descrip

tions as L.C. spaces, which are either inductive or projective limits

or both together of (semi)-inner product spaces.

To this end, we present in this chapter two general types of construc

tions of L.C. spaces. For a directed and separating set p of sequences

and for a countable resolution of the identity on a Hilbert space, we

introduce an inductive limit of Hilbert spaces and a projective limit

of semi-inner product spaces, which are in weak duality. We present a

complete description of the topological properties of those spaces and

the relation between their topological properties and certa:Ïln conditions

on the set p.

The theory, unfolded in this chapter, finds its roots in a University

Report of J. de Graaf, which was published in 1979 .• In this report De

Graaf introduced an inductive limit SX A and a projective limit Tx A' ' .

where X denotes a Hilbert space and A denotes a nonnegative self-adjoint

operator in X. A main feature in this theory is the introduction of a

set of seminorms, which generate the inductive limit topology. See [GJ

for a revised version of the original report.

78 III Inductive and projective limits

In 1981, S.J.L. van Eijndhoven introduced a similar construction, viz.

an inductive limit o(X,A) and a projective limit 1(X,A). See [El).

It turns out that many test spaces and distribution spaces fit in one

of the functional analytic schemes

T(X,A) c;..X c;..o(X,A)

About 1983 it appeared, amongst others, that both inductive limits also

admit a description as a projective limit and, conversely, both projec

tive limits also admit a description as an inductive limit. See [EG 1].

Al this culminated in a theory, which can be found in [EGK]. We give a

rough sketch of the ideas from that paper. Given a set ~ of Borel func

tions on lR, which fulfills certain technical conditions and given a

self-adjoint operator A in X, there are constructe~ an inductive limit

of Hilbert spaces, S~(A)' and a projective limit of semi-inner product

spaces, T~(A)" Here $(A) denotes the collection of operators

{~(A) 1 ~ E ~}. This theory applies to the spaces mentioned in [G],

[E 1], [EG 1]. To each set$ there is associated a set~+ with similar

properties as ~. We mention three important features of the theory.

1. The inductive limit topology is generated by a collection of semi

norms, induced by the operator in ~+(A).

2. The bounded sets in T~(A) have a characterization in terms of the

operators in ~+(A) and the bounded subsets of X.

3. Under a so-called 'symmetry condition' it appears that S~(A)

and T~(A) = S~+(A) as topological vector spaces.

Other developments in more algebraic settings can be found in [EK],

where GB*-algebras play ·a fundamental role, ·and in [tE], where commuta

tive groups and sets with an L1-convolution over these groups lie at

the basis.

The conditions, which are imposed on the set <11,i imply. that the collec

tions of operators ~(A) can be replaced by more simple sets of operators

of the form {LnEIN a(n)XQ(n)(A) Ia€ p}. Hei::e the Q(n)'s are Borel sets

such that the projections XQ(n)(A), n €IN, establish a resolution of

the identity and here p is a collection of sequences with a structure

inherited from the set $.

III Inductive and projective limits 79

The theory in this chapter is essentially the [EGK]-theory, hut it is

looked upon from the following viewpoint: Let (Hm)mEll denote a sequence

of Hilbert spaces with ll a countable set. Put H = emE:ll Hm and let

(Pm)me::D:. d:enote the resolution of the identity operator in H such that

PmH = Hm. We consider inductive limits îf p;(Hm)mE:ll] and projective

limits P [p;(Hm)mEli] originating from the collection of operators

{~lI a(m)Pm 1 a E p}. Here the set pis a directed separating sequence

set.

The first section deals with sets p of nonnegative sequences on ll. ln

the collection of these sets p we introduce a quasi-ordering, a classi

fication and also a cross operation which associates to each set p a

set p# of nonnegative sequences.

In the second section we introduce semi-inner product spaces related to

nonnegative sequences. These semi-inner product spaces are the building

blocks for the main subject of study in this section, namely the induc

tive limits I[p; (H) C1T J of Hilbert spaces and the projective limits m m"'.u.

P[p; (Hm)mEIT] of semi-inner product spaces. Our approach to the induc-

tive limit topology differs from the approach in [EGK].

In the third section we present the so-called cross symmetry condition,

viz. p## ~ p, adapted from [EGK], which implies that the inductive

(projective) limits are projective (inductive) limits.

In an appendix we connect our results with [G] and [EGK].

§ 1. Positive sequence sets

+ . Let Il denote a countable set and let w (Il) be the set of all positive

sequences, i.e.

(1.1) w+(Jr.) = {u E w(:JI) VjEl: u(j) E IR and u(j) ~ O}.

In this section we want to study a specific class of subsets of w+(li).

We use the concepts and notations from Section I.3.


(1.2) For each subset 't of w(lL) let ,+ be defined by ,+ T n w+(:U:). + + + Then we get <P (ll), R,p (][), c0 (ll) and so on.

In w+ (Il.) a partial ordering ::i and a quasi-ordering ~are defined as

follows:

1. 1. Definition

Let a,b € w+ (n:).

Then a ::i b if vj€ll : a(j) ::i b(j)

and a ;s b if 3À>O : a ::i ;>,.b •

Of course, a ::i b and b ::i a imply a = b. However, a ::;, b and b ~ a do not

imply a = b. So the quasi-ordering is certainly not symmetrie. The

relation :f:; leads to an equivalence relation between nonnegative se-

quences.

1. 2. Definition

Let a,b € w+(JI).

The sequences a and b are said to be aeymptotically equivalent if

a :!:, b and b ~ a. If a and b are asymptotically equivalent, we write

a,..., b.

It can easily be checked that ,..., is an equivalence relation.

Next we consider subsets in w + (JI). Both the partial ordering ::i and

the quasi-ordering, ;s in w+(U) induce a quasi-ordering between the

subsets of w + (U).

1 • 3. Definition

Let p,a c: w+ (1I). We deÜne

p s a if vae:p 3be:o a ::i b

P < a ,..., if va€p 3b€a a ':f. b

P R1 a if p ::i a and a ;;; P •

p~a if p~aand a :::, P


(1.3) The following notions are often used. A set p c:w+(X) is called

aeparating if "mEU 3aEp ~ a(m) /: 0 '

con ic if VÀ>O : Àp c: p '

aolid if vaEp vbEw+ (lI) b::ia•bEp

dir>ected if va,bEp 3cEp a ;;; c and b::i c

quaai-dir>ected if Va ,b€p 3c€p a ;;, c and b :$_ c

We remark that p is a solid cone iff p = p • i:(lI). + + + + The sequence sets q> (U), w (U), c0 (U) and R,p(I[), p = oo,1,2, ••• , are

separating solid directed cones.

We present a classification of the subsets of w+(U), as introduced by

A.B.J. Kuylaars in bis Master's thesis [Ku).

1. ll. Definition

Let p c: w + (X).

The set pis type I if there exists ·a finite set cr c: w+(lr.) such that

P ,..., cr.

The set p is type II if p is not type I and if there exists a countable

set o c: w+ (U) such that p "' o.

The set p is type 111 if p is neither type I nor type II.

EXAMPLES:

The set R,:(x) is type I, because 1:(u),..., {11}. The set q>+(U) is type II,

because q>+ (lr.) is not type I and q>+ (U) "' {XE 1 q E lN} where (U ) ElN + q q q

is an exhaustion of ][. The set w (U) is type III. This statement can

be proved by a diagonal argument showing that w+ (:il:) is not type I or

type II.

The main topic of this section is the #-operation on subsets of w+(:Ir.).

1. 5. Definition Let p c: w + (U) •

The sequence set p#, called 'p cross', is ~efined by

82 II! Inductive and projective limits

For each p c w +(Il) the set p# is a separating directed solid cone. It

is straightforward that

pc p## and

+(..,.) h . 1· . 1· # # For p,o E w- J.L t e inequa ity p ;S o imp ies o c p •

EXAMPLES:

[t/Cn;)J# = w+(n) ,

[w+(n)]# =tp+(n) ,

+ # [c0

(n)J

[R,+ (Il)]# ·p

p oo,1,2,". ,

Proofs are left to the reader.

The cross operation leads to the notion of #-symmetry.

1. 6. Definition

Let pc w+(n).

Th . # t . i"f ## e set p is -symme :r>ic p ~ p

From the previous examples it follows that the sets tp+(li), w+(li) and

il,+ (Il) are #-symmetrie, the sets e +0

(n) and il,+ (n), p = 1 ,2, .•. , are not 00 p

#-symmetrie.

We mention some theorems formulated and proved in [Ku], Chapter IV.

1 . 7. Theorem +

Let p cw (Il) be a type II, #-symmetrie set.

h # . II T en p is type I.

1 • 8. Theorem

Let p c w+ (Il) be type I or type II.

Then p is #-symmetrie iff p is separating and quasi-directed.

1. 9. Theorem

Let p, o c w+ (n) be two type II, #-symmetrie sets.

Then (p • o) # ~ p# • o# and (p# • o#) # ~ p • o.


Proof:

We give a proof of the first statement, which is shorter than the proof

of the corresponding statement in UKu]. . . b . h # #. ( )# It it:.is o vious t at p •a ·::,, p•a •

Let u E (p • a)#. We have to prove that;there exists v E p# and w E a#,

such that u ;;; v • w. Let p "' {a 1 n E lN} and a "' {b 1 n E lN} with n · n

a ~a 1

andb :;;b+1,nElN. Put À =max{t,lu•an·bm[

00), n n+ n n n,m

n,m E 1N. Then À :;; À •À for all n,m E 1N. For j E ][ we have n,m n,n m,m

u(j) < inf {1 a-1(j)b-1(j) ·r j E Il[a] and J. E ][[b J} ~ • ·n,m n m n m

;;;inf{À a-1(j)•À b- 1(j)ijE1I[a] andjElI[b]}~ n,n n m,m m n m

j E 1I [a ]} • n

j E ][ [b ]} • m

We define v and WE w+(l[) by

v (j)

w(j) inf{À b- 1(') m,m m J j E 1I •

Then v € l, w € o# and u ~ v • w.

1. 10. Theorem + +"' Let p c w (lI) and a c w (:n::) denote two type II, #-symmetrie sets.

Then (p © o)#,...., p# .© a# and (p# © o#)#,...., p © a.

Finally we introduce the notions of Köthe set and moulding set. Bach

Köthe set yields an inductive limit and a projective limit of weighted

i 2-spaces. A moulding set is a Köthe set of a special type. The defini

tion is taken such that the corresponding inductive limits and projec

tive limits have manageable topological properties. But first we intro

duce multipliers.

a

84 !II Inductive and projective limits

1. 11. Definition Let pc: w+(l[).

A sequence C: is called a p-multiplieP if C: • p ,.., p.

1.12. Lemma

Let ë; be a p-multiplier for some pc: w+(Il). Then

a. The sequence C:-l is a p-multiplier.

b. If :O:[l;] =,Il, then ë'; and C:- 1 are alsi> p 11-multipliers.

Proof:

a. For each a € p we have 1I [a] c: 1I [ë';] • -1 Let a € p. There exists b E p with <: • a ;s b, whence a :S l; b.

Conversely, let c E p. Then there exists d € p with c ;s l; • d and

therefote <;-1 c :., d. So p ..... <;-1 • p.

b. let ll[l,;] =ll. Then (ë'; • p)# = <;-1 • p11• Since l; • p,..., p, we find that ?-l # # Th 7 -l . # 1 . i· S . .,, • p ,..., p • e sequence .., is a p -mu tip ler. tatement a. im-

plies that <: is a p#-multiplier. o

1. 13. Definition

A sequence set p in w+(ll) is a Köthe set if pis separating and quasi

directed. + A Köthe set pis a moulding set if p admits a p-multiplier <:in t1

(:n:).

1.111. Lemma

Let p denote a moulding set.

Then p# is a moulding set,

Proof:

Let <:: denote a p-multipl_ier with <; E t~ (Il.). Since p is separating, we

get Il[{) =Il. From Lemma 1.12 it follows that <:is a p#-multiplier. o

Our concept of Köthe sets is a modification of the one in [KG].


§ 2. Hilbertian dual systems of inductive limits and projective limits

In this section we introduce dual systems of inductive limits and pro

jective limits. Here we feel inspired by Van Eijndhoven and De Graaf,

[EG 1 l, [EG 2] and [E 2]. The presentation here differs fran the presenta

tion in the fore-mentioned references in order to make the topological

structure of our inductive limits and projective limits more explicit.

Throughout the remaining part of this chapter a f ixed sequence of non

trivial Hilbert spaces (Hm)mE:n: is considered. Here n: denotes a count

able set again.

2. 1 • Defini tion

Let XHk denote the Cartesian product k~ ~ endowed with the product

topology.

Elements h of XHk are mappings on :n: with h(m) E Hm' m E Il, and are

occasionally denoted by (h(m)) E.,,.. Fix m E 1[. We identify h E H m,,.. m m with (h o ) En: € XHk. So the Hilbert space H is considered as a ro m,n n m subspace of XHk.

The projection P : XHk + H is defined by P h = (h(m)ó . ) E:D:' h E XHk. m m m m,n n

Let H denote the Hilbert space {h E XHk 1 ~Il llPm hll~ < oo} with inner

product (h,g)H = Î-mEn: (Pm h,Pm g)lfui.

The operators Pml H' m E 1I are mutually orthogonal projections on H and

we have Î-m€1[ Pm h = h, h E H. So H = 4Ï\€1[ H k' Note that the operators

Pm, m E Il, are continuous on XHk.

2.2. Definition

Let a € w + (E).

The linear operator Aa: XHk + XHk is defined by Aa = ~lI a (m)Pm.

Let a,b € t/(u) with a :;;; b. The operators AalH and Abltt are positive

self-adjoint operators in H and i.satisfy the operator inequality


Two types of seminonns will be used. These seminorms are defined on

subsets of XHk.

2. 3. Definition

Let a E </ (lI).

The seminonns Pa and qa are def ined by

Dom(pa) = {h E XHk 1 Aa h E H} ,

\ ! Pa (h) = 111\a hl!H = (~JI lla(m) Pm hll~) , h E Dom(pa) ,

and

q (h) = I lla.(m) P hllH , h E Dom(qa) • a mElI m

2.4. Lemma

Proof:

Let h E Dom(Pi,)· Using Hölder's inequality we get

Î Ha(m)P hllH= I (a·b-1)(m)llb(m)P hllH~

mElI m mEl! m

We recall tbat T(V,pr) denotes the topology on a space V, which is

generated by a collection Pr of seminorms on V.

0


2. s. Corollary

Let V denote a subspace of XHk' Let p c w +(ID) be a directed set such -1 +

that V c n Dom(p) and v E 3bE : :n:[a] c Il[b] and a • b E Jl2(Il). aEp a a p p

Then V c n Dom(q ) and the topologies T(V,q ) and T(V,p ) are the aEp a P P

same.

Proof:

From Lemma 2.4 it fellows V c n Dom(4 ). So for each a E p there aEp a

exists b € p such that

Since p is directed, the collections of seminorms p and q are directed p p

too. Because of Theorem I.1.4 the above mentioned topologies are the

same.

After these preliminary results we are ready to introduce inductive

limits. Building blocks for these limits are Hilbert spaces.

2.6. Definition

LetaEw+(D:).

The Hilbert space H[a] is defined to be the space Aa H endowed with the

inner product

(h,g)H[a] (A -1 h,A _lg)H • a a

h,g E Aa H •

2. 7. Lemma + Let a,b E w (Il), The following results are equivalent:

a. a ~ b •

c. H[a] c;..H[b] •

Proof:

c. "b. This implication is trivial.

-1 b. "a. Statement b. implies Ab_1 H c H. Hence b • a E R.

00(1[) or

•a equivalently a ~ b.

Cl


a. "c. For h E H[a] we have

-1 lhUH[b] = HA _1 hllH;:;; lb • aL,, • HhllH[a] •

b

So H[a] c: H[b] and the canonical injection is continuous.

So fora Köthe set pc w+(1I), i.e. a separating and quasi-directed set

p, the collection {H[a] 1 a E p} is an inductive system. Hence

2. 8. Deflnition Let pc w+(R) be a Köthe set.

The inductive limit 1 [p; (Hk)kER] is defined by

In this chapter we denote I [p; (Hk)kE1I] by I [p;Hk].

2. 9. Theorem + Let p,a cw (1I) be Köthe sets,

Proof:

a. Let p <·a. By lemma 2.7: U H[a] c: bU H[b]. Let j denote the ~ aEp Ep

canonical injection from I[p;Hk] into I[a;Hk]. Let a E p. There

exists b € a such that a ~ b, whence jlH[a] is a continuous mapping

from H[a] into H[b]. Since H[b] c:;...l[o;Hk]' the mapping jltt[a] is

continuous from H[a] into I[a;Hk]. Theorem I.2.3 implies that j is

continuous from I[p;Hk] into I[cr;Hk].

b. Let p ~o. Then p ;[;, a and a ~p. Apply statement a. twice.

Next we consider some topological properties of the inductive limits

l[p;"itl· For definitions see Section 1 in Chapter I.

0

c

III Inductive and projective limits

2.10. Theorem

Let p € w+(JI) be a Köthe set.

a. The space I[p;Hk] is barreled.

b. The space I[p;Hk] is bornological.

Proof:

a, Let W denote a barrel in 1[p;Hk]' i.e. W is a closed, convex, ab

sorbing and balanced subset. Fix a € p. The set W n H[a] is closed

in H(a], whence W n H[a] is a barrel in H[a]. Since each Banach

space is barreled (cf. [Sch], Ch. II, Section 7), the set W n H[a]

is a neighbourhood of o in H[a]. Hence W is a neighbourhood of o in

I[p;Hk]. Cf. the remark after Definition I.2.2.

b. Let W denote a balanced convex subset of I[p;Hk] which absorbs

every bounded set. Fix a € p. Let B denote the open unit ball in H. Since Aa B is a bounded set, the set W ó H[a] is a neighbourhood of

89

o in H[a]. Hence W is a neighbourhood of o. o

In general it is not easy to find explicit expressions for the semi

norms, which generate the inductive limit topology of I[p;Hk]. However,

if we restrict to moulding sets p, i.e. KÖthe sets with l; • p "' p for

some l; € t7(JI), it is possible to describe the inductive limit topology

in terms of explicit seminorms. At this point the sequence sets p# ap

pear on the stage.

+ Let p € w (lI) denote a KÖthe set. For all

have pb(Aa h) = llAa•b hllH:::: la• bl 00 • llhllw

If p • p# c: Q,~ (JI), then i t is obvious that

a € p, b € p# and h € H we

So I[p;Hk] c::: Dom(pb), b € p#. # I[p;Hk] c: Dom(qb), b € p •

The topological considerations are based on the next lemma.

2.11. Lemma # + Let p denote a KÖthe set such that p • p c: t

1 (lI). Let V denote a

balanced convex subset of I[p;Hk]' such that V n H[a] is a neighbour

hood of o in H[a] for each a € p.

Tben there exists U € p# such that


Proof:

Let kv denote the gauge of V. For all h E I[p;Hk] the inequality

kv(h) < 1 implies that h E V. Since V n H[a], a E p, is a neighbourhood

of o in H[a}, there exists µa > 0 such that

For all m € ll, a € p and h € Hm n H[a} we have

UhBH = a(m)llhllH[a] •

Since pis separating, the restriction kvlfiui is continuous at o for

each m € ll. So, we can define the sequence u € w+ (ll.) by

u(m) = sup {kv(h) 1 h E Hm' llhllH = 1} •

We prove that u E p#. Let a E p. Then for all m E lI

a(m)u(m) ;;> sup {a(m)kv(h)/llhl!H 1 h € Hm' h ;. O} ;;>

;;> sup {kv(Aa h)/llAa hHH[a] 1 h E H, Aa h ;. O} ;;> µa •

Hence u E p#.

Let h € I[p;Hk] with qu (h) = ~Ell u(m) llPm hllH < 1. We will show that

h E V. Let h € H[a] for some a E p. Since the seminorm kviH[a] is

continuous at o, i t is continuous on H[a]. The series ~Ell Pm h con

verges to h in H[a]-sense, hence

kv.<h} ;;> i: kv(P h) ;;> I u(m)llPm hllH < t • mE:lI m m€][

So h E V.

2.12. Theorem

Let pc: w+(ll) be a Köthe set.

a. Then T2 = T( U H[a],p #) is contained in TI, the inductive limit aEP P

topology of I(p;Hk]' and so I[p;Hk] is a Hausdorff space.

b. lf in addition p • p# c: ~~ (JI}, then T I is contained in

T1 = T(a~p H[a],qp#) and so T2 c: Tz c: T1•

c


Proof:

a, Let u E p#. For all a E p the seminorm Pu IH[a] is continuous. So

Theorem I.2.3 implies that A .is continuous with respect to the . u

inductive limit topology T1.

91

b. Let p • p# c i7 (lI). Let V denote an open, convex and balanced neigh

bourhood of o in I[p; f\.;1· Hence V 11 H[a] is a neighbourhood of o in

H[a] for each a E p. Because of Lemma 2.11, there exists u E p#,

such that {h E I[p; \1 1 <Zu (h) < 1} c: V. Hence V is a neighbourhood

of o with respect to T1

•

Moulding sets p c w+ (E) have the property that (J • p# c R-7 (E).

2. 13. Corollary

Let pc: w+(E) be a moulding set. Let T2 , T1

and T1 be defined as in

Theorem 2.12.

Then we have T2 " T 1 "" T1•

Proof:

Since p • p# c Q,7 (n:) we have T2

c T I c T1• By assumption there exists

a p-multiplier C: E i7 (n:). For each a E p there exists b E p such that

C1 • a :S b, whence ll:[a] cE[b] and a • b-t :S (. Corollary 2.5 yields

that r 2 " T1, whence the three mentioned topologies are equal.

The previous corollary can be extended. A Köthe set p, which satisfies

3 v 3 : C 1 • a ::., b • <:n7 (lI) ,n.[<:1 =lI aEp bEp

is in fact a moulding set (with p-multiplier Ï:).

For a Köthe set p, which satisfies the weaker condition

# + we also have that p • p c i 1 (Il.) and that Corollary 2. 5 implies that

the three topologies, mentioned in Theorem 2.12, are equal.

We state three more results on the inductive limits I[p;Hk] where p is

a moulding set.

a

0

92 !II Inductive and projective limits

2.111. Theorem + Let p,crc: w (II) be moulding sets. The following statements are equiva-

lent:

a. 1 [p;Hk] c;. 1 [cr;Hk] •

b. I[p;Hk] c: I[cr;Hk) as sets.

c. p ~ a •

Proof:

a. • b. This implication is trivial.

b. • c. Let <: E R.1 (II) denote a p··multiplier. Let a E p and let h E XHk

with llPm hllH = 1, m E lI. Then Aa(h) = Az:-1.a<Az: h) E I[p;Hk). Hence

there exist b E cr and g E H. such that Aa(h) = Ab(g). So for all

m E II

a(m) = b(m)llPm gllH ,

whence a ::, b.

c. "° a. LE!l!lllla 2.9.a. yields this implication.

2.15. Corollary

Let p,cr cw+(II) be moulding sets.

Then I[p;Hkl = I[o;Hk] iff p,..., cr.

The next theorem can be found in a similar form in [Ma 2].

2. 16. Theorem

Let p c: w +(II) be a moulding set and let d (m) = dim(Hm), m E lI.

Then the space I[p;Hk] is nuclear iff

+ VmE][ : d(m) < oo and "aEp "'ue:p#: a • u • d E R.1 (II) •

We remark that the condition a • u • d E i~ (II) is equivalent with the

condition that A ltt is a trace class operator on H. a•u Proof:

Let w E p#. The Hilbert space H[w-1] is a completion of the quotient

space 1 [p;Hk]/p;(o).

0


Now I[p;Hk] is nuclear iff for each u E p# there exists v € p# with the

properties:

(2.1) u ~ v '

(2.2) -1 -1 The canonical injection An : H[v ] c;..H[u ] is nuclear.

u -1

Since A _1 and Au are bounded operators from H into H[v ] and from -1 v

H[u ] into H, respectively, condition (2.2) can.be replaced by

(2.3) The operator Au·v-l is a trace class operator on t/.

Note that trace(J\ _1) = I.' (u • v-1) (m) trace(P ) u•v m~JI m

". Let a E p and u E p# By assumption there exists v E p# such that

u ~ v and Au·v-l is a trace class operator on tl. So

Since p and p# are separating, all d(m) are finite •

.-. Let l; E i7 (JI) denote a p-multiplier. Let U E p#. Put V = d • ë;-1 • u.

Then u;:;, V and the assumption implies that v E p#. Finally we have

I.' -1 trace(A -1) = t. ê;(m)d (m) trace(P ) u•v mEJI m

Next we consider projective limits. We construct these projective

limits from semi-inner product spaces.

2. 17. Definition

Let a E w+(JI).

The semi-inner product space H+[a] is defined to be the vector space

endowed with the semi-inner product

+ h,g E Aa (ff) •

The space tl+[a] is a Hilbert space iff JI[a] =II. In that case

tl+[a] = H[a -l].

0


2.18. Lemma + Let a,b E w (][).

Then a ~ b iff ff+[b] c;.. ff+[a].

Proof:

•· Let h € H+[b]. Then

So ff+[b} c:;.ff+[a].

". There exists y > 0 such that Vh€H""[b] : IAa hi 2 ::á YIAb hi2•

Since H c: H+[b], we find that a(m)';;î yb(m) for all m EU. Soa ;S b. c m

Fora Köthe set pc: w+(lr) the set {H+[a] 1 a € p} is a proji;ctive

system. Cf. Definition I.2.4.

2. 19. Definition

Let pc: w+(1l) be a Köthe set.

The projective limit P[p; (fik) kEII] is defined by

Lemma I.2.6 implies that P[p;Hk] is the space n H+[a] with the topo-

b h • . ae:p . . h

logy generated y t e seminorms Pa• a € p. Since p is separating, t e

space P[p;Hk] is Hausdorff.

2. 20. Theorem + Let p,o c: w (11) be Köthè sets. Then

Proof:

a. Let p < cr. Lemma 2.18 implies n H+[a] c: bn H+[b]. Since for each ~ a€o Ep

a E p there exist y > 0 and b E o such that


we get that P[a;Hk] ~ P[p;Hk].

Conversely, let P[a;Hk] ~ P[p;Hk]. Continuity of the injection

implies that for each a E p there exist y > 0 and b E a such that

Pa(h) ~ ypb(h) for all h E P[a;Hk]. Since Hm c H, m E 11, we find

that a ;;;; yb.

b. We remark that p ~ a iff p :;;, a and a :;;, p.

Next we consider some topological properties of the projective limits

P[p;Hk]. For definition see Section I.1.

2. 21. Theorem + Let p c w (11) be a Köthe set.

The space P[p;~] is complete.

Proof:

The proof is standard. Let (hi)iEJ denote a Cauchy net in P[p;Hk].

For all m E 11, the net (P h.) .EJ is a Cauchy net in H with limit h • m l. l. m m

95

D

Let h E Xflk with P H = h • For each a E p the net (A b.) 'EJ is a Cauchy m m a i i

net and bas limit ha € H. It turns out that ha = Aa h.

Hence h € P[p;Hk] and h is the limit of the Cauchy net (hi)iEJ" a

2. 22. Theo rem

Let pc w+(ll) be a moulding set.

For all b E p we have the inclusion En H+[a] c Dom(qb) and the projec-. a P ,...

tive limit topology of P[p;Hk] is equal to T( n H [a],q ). aEp p

Proof:

The projective limit topology equals T( En H+[a], p ) • Since the set p a P P

is moulding, for each a E p there exists b E p such that a ~ b and

a • b-1 E R-2(11). Corollary 2.5 implies the statement. o

In the following theorem we characterize bounded sets in P[p;Hk].

2. 23. Theorem

Let p cw+(ll) be a moulding set. Then

96 Ill .lnductive and projective limits

a. For all u E p# and all bounded subsets 8 of H the set AU B is bounded

in P(p;Hk].

b. For a bounded subset W in P[p;Hk] there exist u E p# and a bounded

subset B of H such that Au maps B hoineomorphicly ànto W with

respect to the relative topologies.

Proof:

a. Let u E p# a.nd let 8 be a bounded set in H. For all a E p and h E B

we have

So J\u 8 is bounded.

b. Let W be a bounded set in P[p;Hk]. Let ~ E i7(lI) denote a p-multi

plier. For m E lI put

-1 1 u(m) • ~ (m) • sup {llPm hllH h E W} .

Since p is separating, U(m) < oo for all m € Il.

Let a E p. For all m E :U: we have

a •u(m) = sup {i'.;-1 (m)a(m)llPm bllH 1 h E W} ::1

::1 sup {q~-t·a(h) 1 h € W} < oo.

Hence u E p#. Further, for all h E W we have An h = h and for all . u

m E lI llPmAu-1 hllH ::1 ~(m). Put B = {Au-1 h 1 h E W}. Then Bis a

bounded set in H, the set W equals A0

B and J\u: B + W is a bijection.

Naw we prove that Au is a homeomorphism. Fix h E B. For a;n g E B

and a E p we have pa(./\u(h-g)) ::1 la• ul00

• Nh-gllH' So ./\u maps 8 con

tinuously onto W. Finally we prove continuity of ./\u-t from W onto 8. Let h E W. For

all g E W and finite subsets lF of :0:


-1 + Since {u • XIF} ;;;, p and ?.; E t 1 (Il) , we get that 11.u-1 llîaps W con- .

tinuously onto 8.

2. 2~. Corollary

Let pc: w+(E) be a moulding set and let W denote a subset of P[p;Hk].

Then W is compact iff there exist u E p# and 8 c: H, 8 a compact subset,

such that W = 11.u B.

2. 2 5. Corollary

Let p c w +(Il) be a moulding set and let h E XHk.

Then h E P[p;flk] iff there exist u El: p# and g E H such that h = Au g.

So P[p;Hk] = I[l;Hk] as sets.

We recall that a L.C. space in which all bounded and closed subsets are

compact, is called a semi-Montel space.

2. 26. Theorem

Let pc: w+(Il) be a moulding set.

Then the space P[p;Hk] is semi-Montel iff VmEil dim(flm) < 00 , i.e. Pm

is of finite rank.

Proof:

=>. Let B denote the closed unit ball in H • Then B is a bounded and m m m

closed subset of P[p;Hk] and therefore a compact subset of P[p;Hk].

Hence Bm is compact in H and so dim(fl ) < 00• m m

~. Let W denote a bounded and closed subset in P[p;flk]. Let ?.; E i~ (Il),

u E p# and B c: H be the same as in the proof of Theorem 2.23.b. For

all h E 8 and m E :0:, llPm hll H :ii l.;(m). Since all Hm are fini te dimen

sional, the set B is compact. Because 11.u is a homeomorphism from B onto W, the set W is compact,

In the final part of this section we define a pairing between the in

ductive limit J[p;flk] and the projective limit P[p;Hk]. The pairing is

sesquilinear.

2. 27. Definition

Let pc w+(:U:) be a Köthe set.

By a Hilbertian dual system we mean the pair of spaces I[p;Hk] and

0

0


P{p;ffk] with their pairing <•,•>, defined by

with a E p such that h E H[a].

The pairing in Definition 2.27 yields a representation for the duals of

our inductive/projective limits. First, however, we introduce classes

of linear functionals.

2.28. Definition + Let p c: w (lI) be a KÖthe set.

For g E P[P;f\l we define the linear functional ~g on I[p;Hk] by

~g(h) = <h,g>, h E I[p;ffk].

For h E I[p;HkJ we define the linear functional 'l'h on P[p;Hk] by

'l'h(g} = <h,g>, g E P[p;Hk].

For h E H we define the linear functional rh on H by fh(g)

g E ff.

(g, h) H'

2. 29. Theorem + Let p c::w (ll) be a Köthe set. Letland m denote linear functionals on

I[p;Hk] and P[p;Hk]' respectively.

a. The functional l is continuous iff there exists g E P[p;Hk] with

t = ~ • g

b. The functional m is continuous iff there exists h E I[p;Hk] with

m = 'Vb.

Proof:

a. •. Since t o PmlH is continuous, there exists 8ni E Hm such that

l 0 PmlH = r &!' m E ll. Let g E XHk with Pm g = ~· Fixa E p. The restriction lltt[a] and hence l o Aaltt• is continuous.

So there exists fa E H such that to AalH = rfa' We find fa= Aa g.

Bence g E P[p;Hk] and l = ~g· .

""'· Let l = ~g for some g E P[p;Hk].

For all a E p we have


whence llH[a] is continuous. Sol is continuous.

b. "*• Since m is continuous, there exist µ > 0 and a E p such that

Hence m 0 Aa-1 IH is continuous and there exists f E H such that ----

m o Aa-11 H = r f" For g E P[p;Hk] we have m(g) = (Aa g,f)H = <Aa f,g>.

So m = '!' (A f). a

<=. Let m = '!'(Aa f) for some a E p and f E H. For g E P[p;Hk] we have

lm<g) 1 = 1 <A f ,g>I a

So m is continuous.

We finish this section with two theorems on strong duals. Only sketches

of their proofs will be given.

2. 30. Theorem

Let pc w+(Il) be a moulding set.

Then the mapping '!': I[p;Hk] + (P[p;Hk])~ is an antilinear homeomorphism.

Proof:

Theorem 2.29 yields that '!' is an antilinear isomorphism from I[p;Hk]

onto (P[p;Hk])'. The tii>pology of (P[p;Hk])S is equal to

where B is the unit hall in H and where

sup ll<Au h) 1 , hEB

# Now, for each U E p and h E I[p;Hk] we have

a

a

100 111 Inductive and projective limits

2. 31. Theorem

Let pc w+(I) be a moulding set.

Then the mapping <P: P[p##;Hk]-+ (l[p;Hk])S is an antilinear homeomorphism.

Proof:

The proof is similar to the previous one. We remark that

and the topology of (I[p;Hk])~ is equal to

where Be = B n \l>(ID). c

§ 3. Cross symmetrie moulding sets

In this section we investigate which topological conditions on the

spaces I[p;Hk] and P[p;Hk] are equivalent with the #-symmetry condition

on the moulding set p.

3. 1 . Theorem

Let pc w+(I) be a moulding set.


a. The set p is #-symmetrie.

b. I[p;Hk] ##

I[p ;Hk]

c. P[p;Hk] ##

P[p ;Hk]

d. I[p;Hk] #

P[p ;Hk]

P[p;Hk] # e. I [p ;Hk]

Proof:

a. • b. See Corollary 2.15.

a. • c. See Lemma 2.20.b.


a. * d. Observe that p# = p###. Corollaries 2.13 and 2.25 yield that

P[p#;Hk] = l[p##;Hk].

101

# ## a. * e. Corollaries 2.13 and 2.25 yield that l[p ;Hk]=P[p ;Hk]. a

For the notions in the second theorem we recall Section I.1.

3. 2. Theorem

Let p E w+(JI.) be a moulding set.


a. p is #-symmetrie.

b. P[p;Hk] is reflexive.

c. I[p;Hkl is reflexive.

d. P[p;Hk] is barre led.

e. P[p;Hk] is bornological.

f. I[p;Hk] is complete.

Proof:

a. * b. From Theorems 2.30 and 2.31 it follows that the spaces ## ( ') 1 • ## P[p ;Hk] and (P[p;Hk])S S are homeomorphic. Hence P[p ;Hkl

= P[p;Hk] iff P[p;Hk] is reflexive.

a. * c. Since (P[p;HkDé = (P[p##;HkD~" Theorems 2.30 and 2.31 imply

that the spaces I[p##;Hk] and ((I[p;Hk])S)~ are homeomorphic. Hence

l[p##;Hk] = I[p;Hk] iff I[p;Hk] is reflexive.

a • .,. d.,e. By Theorem 3.1.e. and Theorem 2.10 these implications follow.

a • .,. f. By Theorem 3.1.d. and Theorem 2.21 this implication fellows.

d. "*' a. Let a E p##, Let W denote the set

{h E P[p;H ] 1 sup {a(m)llP hllH 1 m E JI.} ::l 1} m m

Then W is a barrel, because W is a closed, convex, balanced and

absorbing set. Hence W is a neighbourhood of o in P[p;Hk]. So there

102 III Inductive and projective· limits

exist µ > 0 and b € p such that

Bence a ~ b.

e ..... a. Let a € p##. Let W = {h € P[p;Hk] J qa(h) < 1}.

Thenlll is a convex and balanced subset of P[p;Hk]. It is easy to see

that W absorbs every bounded subset of P[p;Hk]. Hence W is a neigh

bourhood of o in P[p;HkJ. Further, see the proof of d. • a.

f. ... a. Let a E p#ll, let (Jiq) qElN be an exhaustion of :0: and let g € H.

Then CAa•xn g} €lN is a Cauchy sequence in I [p;Hk] with limit h. q q ##

It appears that h =Aa g. So Aag E I[p;Hk). Hence I[p ;Hk] c

c I[p;Hk] as sets. From Theorem 2.14 it follows that pis #-sym

metrie.

The following theorem deals with the (semi-} Montel property. Cf. Theo

rem 2.26.

3. 3. Theorem

Let p cw+(Iî) be a moulding set.


a. The set p is #-synnnetric and vm€Iî : dim(Hm) < ""·

b. The space 1 [p;~] is semi-Montel.

c. The space Î[p;Hk] is Montel.

d. The space P[p;Hk] is Montel.

Proof:

# a .... b. By Theorem 3.1.d. I[p;Hk] = P[p ;Hk]. Because of Theorem 2.26

the space PCP';Hk] is semi-Montel.

b .... a. Fix m € ll. Let B be the closed 1.mit ball in H • Then B is a m m m closed and bounded set in I[p;Hk]. Hence Bm is compact in Hm' So

dim(ff ) < 00•

m ## Let a € p , let (Irq) qElN be an exhaustion of JI and let g € H. Let

W denote the closure of {Aa·xJI g 1 q € lN} in I[p;Hk]. The set W .. q

c


is bounded and closed. Because <Aa•xnq g) qElN is a Cauchy sequence

in W. the element Aa g belongs to (tl c 1 [p;~]· Hence I [p##;Hk] c:

c I[p;Hk] as sets. From Theorem 2.14 it follows that pis #-sym

metrie.

a. * c. This equivalence follows easily from a, * b. and from Theorem

3.2.

a. * d. This equivalence fellows from Theorems 2.26 and 3.2.

The last theorem deals with nuelearity.

3. ll. Theorem

Let pc w+(Il) denote a moulding set, whieh is #-symmetrie.

Then I[p;Hk] is nuelear iff P[p;Hk] is nuelear.

Proof:

Sinee p is #-symmetrie, P[p;Hk] "' 1 [p# ;Hk]. The remaining part of the

103

Cl

proof consists of an application of Theorem 2.16. c

Appendix

In this appendix we want to indicate how the theories [G] and [EGK] fit

in our theory. The spaces introduced in both these papers will he de

seribed in terms of the theory as developed here.

In [G] the spaees SH,A and TH,A are introduced, A is an unbounded,

nonnegative self-adjoint operator in a Hilhert space H. The analyticity space SH,A is defined by

lim ind e -tA(H) t > 0

where e-tA(H) is regarded as a Hilbert space with inner product -tA tA tA

(e h,e g)H, h,g E e (H).

The trajectory space TH;A is the space of all mappings F: (O,oo} -* H, whieh satisfy

(A.1) Vt,t>O e-tA F(t) "'F(t+ T) ,


endowed with the topology generated by the seminorms F >+- llF(t)llW

F € TH A' where t > O. • +

The sets 'PG and 'PG of functions on 1R are defined by

-tÀ 'Î!G = {À >+- e • À € 1R 1 t > O}

and

'Î!+ is the set of all nonnegative Borel functions f on lR, G

such that for all <.P € \t>G sup <,p(À) f(À) < "'. À€1R

Central results in (G] are:

+ (A.2) The seminoms x i+ llf(A)xllH' x € SH,A' f € \t>G' generate the

inductive limit topology. Cf. [G], Part B, The0rem 1.4.

(A.3) For each F € TH A there exist f € IP~ and b E H sucb tbat -tA ' F(t) = f(A)e h, t > O. Cf. [G], Part B, Tbeorem 2.3.

Now we give a description of the spaces SH,A and TH,A in our terminology.

Let E denote the spectral resolution of the self-adjoint operator A.

Put lI = {m € JN+ 1 E((m-1,m]) /< O}. For each m € IL let Pm denote the

orthogonal projection E((m-1,m]) and let H denote the Hilbert space m

Pm H with induced inner product. So H = em.Ell: Hm. + For t > 0 we define the sequence et E w (:n:) by et(m) -tm

e · m € rr. Put Pe = {et t > O}.

(A.4) The set Pe is a Kötbe set and even a moulding set, because the -1 sequence m o+ (mt + 1) , m E lI, is a Pe-multiplier in R-1(lI).

(A.5) Since Pe,..., {e(l/n) 1 n € JN}, the set Peis type II and #-sym

metrie. Cf. Theorem n1:1. 8.

Fix t > 0. We have

(A.6) v v me:rr À€(m-1,m]

These inequalities imply that e-tA(H) = H[et]. Cf. Definition 2.6.

So, in the terminology of this cbapter


Let m € lI and F E Ttt,A• Then, for all t > 0, etA Pm(F(t))

due to the semigroup property (A.1).

eA P (F(1)) 111

The inequalities in (A.4) imply that the mapping F 1+ (eA P111

(F(1)))m.EJ[

is a homeomorphism from T H,A onto P[p; (ff111

)111

€1[] •

We remark that (A.2) corresponds to Corollary 2.13 and (A.3) is a

weaker version of Theorem 2.23.

Because of the properties (A.4) and (A.5) of the set pe• all results in

[G] can be derived from the theory in Sections 2 and 3. An elaborated

and extended version of the'ideas in [G] can be found in (EG 1].

The theory in [G] has been developed into the theory presented in [EGK].

We give a sketch of the spaces S~(A) and T~(A) introduced in the latter

paper. For this, two ingredients are needed:

(A.7) The first ingredient is a set ~. which consists of nonnegative

Borel functions on IRn, bounded on bounded sets and which has the fol

lowing properties

0. The set ~ is quasi-directed, i.e.

1. For each qi E ~the function qi-l defined by

-1 -1 f (q>(À)) '

cp (;\) "' 0 ,

is bounded on bounded sets.

qi(J..) "" 0

cp(À) = 0

2. The supports ~ = {À € IRn ! q>(À) ~ O} of <P, cp E ~. cover the whole

IRn: IRn= U q>. <PE~ -

3. <P(À) :ll y sup ÀtQ(m)

1/J(À) •


(A.8) The second ingredient is a collection of n strongly commuting

self-adjoint operators, A1, ••• ,A, on a Hilbert space H. It means tbat . n the corresponding spectral projections of the operators Ait• 1 ~ k ~ n,

mutually commute.

Now, let ek denote the spectral resolution of the self-adjoint operator

Ait• k = 1, ••. ,n. Let e denote the joint spectral resolution correspond

ing to the n-set A1, ••• ,An; we mean that eis a projection valued

measure on the o-algebra B(IRn) of Borel sets in IRn, satisfying for all

Borel sets fk in IR, 1 ~ k ~ n,

Let Bb(IRn) denote the set of all bounded Borel sets in IRn.

All spaces introduced in [EGK] are contained in the so-called 'spectra!

trajectory space' G+ which consists of all mappings F: Bb(IRn) ~ H with

the property: F(A1

n A2

) = e(A2

)F(A1), A

1,A

2 E Bb(IRn).

The Hilbert space emb(H) c: G+ is the space of all spectra! trajectories

Fh: A 1+ e(A)h, h E H, endowed with the inner product (Fh,Fg)emb(H) =

= (h,g)H. Naturally, emb(H) and H may be identified.

For q> € ~ the Hilbert space q>(A) • H is the space of all trajectories

F h: A 1+q>(A)e(A n q>)h, h € H, endowed with the inner product q>, . -(Fq>,h'F<P,g)q>(A)•H = (e(~)h,g)H.

Due to the properties of ~. the collection of Hilbert spaces

{q>(A)•ff 1 q> € ~} is an inductive system.

The inductive limit S~(A) c: G+ is defined by

S~(A) = lim ind q>(A)•H • q>E~

+ The space T~(A) c: G is defined to be the space

{F € G+ 1 q>(A)F € emb(H)} ,


endowed with the topology generated by the seminorms F i+ ll1P(A)Fllemb(H)'

F € T <fî(A) , where IP € .

We remark that the spaces SH,A and TH,A can be identified with the

spaces S (A) and T (A)' respectively. G G +

To the set there is associated a set •

(A.9) The set + consists of all nonnegative Borel functions f which

satisfy:

- The function f- 1 is bounded on bounded sets,

- Y E : sup f (À) (fl(À) < '" • IP À€lRn

The set + satisfies the properties O. to 3. in (A.7). Cf. (EGK], Lemma

1 .5.

In [EGK] important results are the following:

+ (A.10) The seminorms F t+ llf(A)Fllemb(H)' F E S(A)' where f € , gener-

ate the inductive limit topology. Cf. [EGK], Theorem 1.7.

(A.11) A set V c T(A) is bounded iff there exists f E + and a bounded

,set Bin emb(H), such that V • f(A)B. Cf. [EGK], Theorem 2.4.II.

In [EGK] a symmetry condition bas been introduced (axiom A.IV.).

(A.12) A set satisfies axiom A.IV. iff for each IP E ++ there exist

Ijl E and y > 0 such that YÀElRn : 1P(À) :;; ylji(À).

We mention an important res'ult:

(A.13) If the set satisfies axiom A.IV., then we have the following

equalities as topological vector spaces:

We now present a description of the spaces S(A) and T(A) in our

terminology.


Let 1l = {m € 71.n 1 t(C\n) f: O}.

For m € 1l let Hm denote the Hilbert space t(C\n)ff,

The mapping F i+ (F(Q(m)) )m€lI is a bijection from G+ on to mh ffm.

For q> € 41 the sequence d € ut (lI) is defined by d (m) = sup q>Ü,), IP IP À€Q(m)

m € ll. Let p41 = {dq> 1 q> € 41} •

(A. 14) The set p41

c:: 1./ (lI) is a moulding set. The sequence <; € R.1

(lI)

with ë,;(m) = (1 + lml1)-(n+1), m € lI, is a p

41-multiplier.

(A.16) If 41 satisfies axiom A.IV., then the set p41

is #-symmetrie.

Translated in our terminology, we conclude that

The result (A.10) on the inductive limit topology of S4l(A) corresponds

to Corollary 2.13 and the result (A.11) on the bounded sets in T4l(A)

corresponds to Theorem 2.23. The result (A.13) bas been a source of

inspiration for Section 3.

We finish with a technical remark and a genera! remark.

(A.17) The set p41

is #-symmetrie iff for all q> E 41++ there exist y > 0

and 1fi E 41 such that

VÀE U Q(m) : q>(À) ~ Yifi(À) • mE:a:

In fact, the condition that p41 is #-symmetrie is weaker than axiom A.IV.

(A.18) Our theory on the spaces I[p;(ffm)mElI] and P[p;(ffm)m€lI] for

moulding sets p can also be described in the terminology of the theory

in [EGK]. This description is highly nonunique, because there is free

dom in the choice of 41 and of A1, ••• ,An.

CHAPTER IV

SPACES OF ANALYTIC FUNCTIONS ON SEQUENCE SPACES

Introduction

The theory presented in Chapter II and in Chapter III culminates in

this final chapter. With the aid of reproducing kemel theory we intro

duce spaces of analytic functions on sequence spaces, which admit an

I-space structure or a P-space structure. Most ideas and tecbniques in

this final chapter appear to be new.

Let us first describe the sequence spaces, on which the analytic func

tions are defined. For a countable set ID each a € w+ (ID) fixes the

Hilbert space R.2[a;ID] and the semi-inner product space J1,;[a;ID]. Here

R.2

[a;ID] is the space a • R.2 (ID) endowed with the inner product -1 -1 + -1 (x,y) 1+ (a •x,a •y) 2 , and J1,2 [a;ID] is the space a • J1,

2[ID] + Cla • w(ID)

endowed with the semi-inner product (x,y) ,... (a•x,a•y)2

• Clearly, for a

Köthe set g c: w+(ID) the Hilbert spaces J1,2

[a;ID], a Eg, establish an

inductive system generating

Gind[g] "' lim ind J1,2[a;ID] a E g

+ and the spaces J1,2[a;ID], a Eg, establish a projective system generating

Gproj[g) lim proj J1,;[a;ID] • a E g

These spaces have been introduced in [E 2] and fit in the scheme of

Chapter III.

Now we want to introduce function spaces on G. d[g] and G .[g]. Our l.ll prOJ

candidates are:

Fproj [g] = n {F ! F1J1,z[a;ID) E S(J!,2 [a;ID])} , aEg

= u {F 1 F E ea (s<R-2(ID)))}. aEg

110 IV Spaces of analytic functions

We remark that Sa F = F(a•.) and that S(R,;[a;ID]) is defined to be

ea (s <R.2 (ID))). Anybow, F . [g] and F. d [g] are subspaces of A(<,p(ID)). proJ in

The main part of this chapter is devoted to a thorough introduction of

the above spaces F .[g] and Fi.nd[g]. Remarkably, the structure of prOJ these spaces turns out to be very similar to the structure of the

spaces introduced in Chapter III.

Let us explain this a bit. Each a E w+(ID) induces two subspaces of

A(<,p(ID)). The space Find[a] = H(<,p(ID),Ka) with Ka(x,y) = exp(a•x,a•y) 2 and the space F . [a] consisting of all F E A(<,p(ID)) with prOJ ea F E H(<,p(ID),KU). F. d[a] and F .[a] are called elementary spaces. in proJ Next we consider inductive and projective systems, induced by these

elementary spaces. Therefore we involve Köthe sets g, which are conic,

and introduce the compound spaces F. d[g] and F .[g]: in proJ

lim ind F. d[a] aEg rn

lim proj F .[a]. a E g proJ

Initially, the elements of F. d[g] and F .[g] are analytic on <,p(ID). in prOJ However, they admit a larger domain. Indeed, since each element F in

Find[a] bas a unique extension F E ea(S(J1,2(ID))) to Jl,;{a;ID], Find[g]

can be considered as a space of analytic functions on G .[g]. prOJ Further, each element Fin F .[a] uniquely determines an analytic

A proJ function F. in S(J1.2 [a;ID]). So the space F .[g] can be regarded as s

prOJ space of functions on G. d[g]. These functions are ray-analytic. Howin ever, only for type II sets g we are able to prove that they are ana-

lytic on G. d[g]. in

As expected, the compouQd spaces F. d[g] and F .[g] have a very rich in proJ structure:

To begin with, we obtain spaces of analytic functions on <,p(ID) with a

well specified growth behaviour and surprisingly, all are subalgebras

of A(<.P(ID)).

As L.C. spaces F. d[g] and F .[g] fit in the scheme of the previous in prOJ chapter. The elements of F. d[g] and F .[g] are uniquely fixed by . in prOJ sequences in w(:M(ID)) (lM(ID) is the set of multi-indices over ID).

For this we look at the coefficients, labeled by :M(ID), in the monomial

IV Spaces of analytic functions 111

expansions. Introducing a so-called up operation on the collection of + + +

subsets of w (ID), which links to each g c w (ID) a set up [g] c w (:M(ID)),

we arrive at the following identifications:

Fproj [g] - G . [up[g]] • prOJ

So the sequence space Gind[up[g]] induces a function space on the

sequence space G .[g]; similarly, G .[up[g]] induces a function proJ proJ

space on G. d[g]. 1n

As already observed, the spaces G. d[g] and G .[g] fit in the scheme 1n proJ of Chapter III. The sequence space analogue of the important notion of

moulding set from the previous chapter is the notion of generating set,

leading to a very rich topological structure of the spaces Gind[g] and

G .[g]. If gis a generating cone, then up[g] is a generating set and proJ

the spaces F1.nf[g] and F .[g] have a rich topological structure, too. prOJ

We finish with a short description of the contents of this chapter.

Section 1 deals with generating sets and Section 2.with the sequence

spaces they induce.

Section 3 may be considered as an intermezzo. We complete the descrip

tion of the analytic function spaces A(qi(ID)) and A(w(ID)) and describe

their duals in terms of analytic funcions of exponential growth.

Sections 4 and 5 are devoted to F. d and F .• In Section 4 we present 1n prOJ the elementary spaces F. d[a] and F .[a] and in Section 5 the c01D-

1n · proJ pound spaces F. d [g l and F . [g]. In both sec tions reproducing kemel 1n proJ theory is an essential tool.

As an illustration of the theory developed in this chapter, we describe

representations of the generalized Heisenberg groups

in the spaces F. d[g] and F j [g], respectively (Appendix). 1n pro:


§ 1. Generating sets

In Chapter III, Section 1, we have introduced Köthe sets and moulding

sets. A set pc w+(ll), where JI is a countable set, is a Köthe set if

pis separating and quasi-directed. A Köthe set p.c w+(JI) is a moulding

set if there exists <; € R,~ (E) such that <: • p ,..., p.

In this section we consider generating sets. The notion of generating

set is an extension of the notion of moulding set. A set g c w+(ID),

where lD is a countable set, is a generating set if it is a mixture of

a block structure and a moulding set. This means that there exists a

sequence w € w+(lD) without zeroes, a partition {Q(m) 1 m € ll} of ID

and a moulding set p c w+ (ID) such that

In Section 5 of this chapter we introduce inductive limits F. d[g] and in projective limits Fproj[g] of analytic function spaces with the aid of

conic Köthe sets g c w+(ID). If g is a generating cone, then the topo

logical structure of the spaces F. d[g] and F .[g] is similar to the in proJ topological structure of the inductive limits and projective limits,

induced by a moulding set as in Chapter III.

We start with positive sequence sets with a block structure. In the

sequel ID and :n: denote countable sets.

1. 1 • Definitlon

Let Q(Jî) • {Q(m) 1 m € Jî} be a partition of ID and let p be a subset

of w+ (Jî).

The set p (!) Q(E) in w+ (ID) is defined by

P 0 Q(ll) = ~n c(m)xQ(m) 1 c E p}

So the set p 0 Q(ll) consists of sequences, which are constant on the

blocks Q(m), m €ll. Naturally, properties of the set p carry over to

the set p e Q(Il).

IV Spaces of analytic functions

1.2. Lemma

Let Q(][) be a partition of ID and let p,cr c w+(][) be Köthe sets.

Then the following .statements are valid:

a. p © Q(li) is a Köthe set.

b. p ~ cr iff p E> Q(li) ~ cr E> Q(][) •

c. (p E> Q(D:))#..., /' © Q(D:).

Proof:

We only prove a part of c. We show that (p © Q(D:)) # ~ l © Q(ll).

Let d E (p © Q(li)) 11 • For all c E p we have

whence

So

and

m~'1I C (m)xQ(m) • d E i00 (ID.~ ,

( c(m) sup d(j)) E Jl,00

(][) •

jEQ(m) mm

( sup d(j)) E p# jEQ(m) mE:n:

d:i: I (sup d(j)\jXQ(m)Ep11

@Q(li). mED: jEQ(m)

113

As the following lemma shows, not every parti tion Q(][) of ID fits well

to a Köthe set g c w+ (ID).

1. 3. Lemma + . Let g c w (ID) be a Köthe set and let Q(][) be a partition of ID.

Then there exists a Köthe set p c w+(D:) such that g ..., p © Q(Jl} iff

Proof:

VaEg 3b€g 3À>O VmEll : J.EsuQp(m) a(j) :i: À inf jEQ(m)

b(j) .

.,., Let a Eg. There exist À1,À2 > 0, c E pand b Eg such that

c


Hence we get

VmE1l : sup a(j) :ii À1 c (m) ;;; À2 inf b(j) • j€Q(m) jEQ(m)

+ •• For each a € g the sequence ;1 € W (][) with ;1 (m) "' SUI> a (j). . . d . jEQ(m)

m € 1l, is well-define because of the assumption.

Put p "' {a E w + (JI) 1 a E g}. Since g is a KÖthe set, p is a Köthe

set. By assumption, for each a E g there exist À > 0 and b E g such

that

Hence g...., p@ Q(Il).

1. 4. Corollary

Let g c w+(ID) be a Köthe set, let Q(lI) be a partition of ID and let + p c w (lI) such that g "' p 0 Q{lI).

For each a E g put a = (. sup ) a (j) )mE1l and put g = {a 1 a E g}. + . JEQ~m

Then g c w (1l') and p ...., g.

Proof:

By Lemma 1.3 it follows that g c w+(lI). From the proof of Lemma 1.3 it

follows that g "'g 0 Q(lI). Lemma 1.2.b. implies that p "'g. c

Now we introduce the notion of generating set.

1. 5. Definition + Let g c: w (ID).

The set g is called a gener>ating set if thete exist a sequence w € w+(ID)

without zeroes, a partition Q(Il) of ID and a moulding set pc w+(Il) -1 such that w •, g ,.., p 0 Q_(1l).

The triple (w,p,Q(Il)) is called a skeleton of g and in particular the

sequence W is called a wei9ht sequenae of g.

REMARKS

(1.1) Generating sets are Köthe sets.

(1.2) For each sequence w E w+(JD) with full support, for each parti

tion Q(1l) of ID and each moulding set p c: (/ (lI), the set w • (p t!> Q(Il))

IV Spaces of ana~ytic functions

is a generating set. In particular, a moulding set p c (/(1I) is a

generating set with skeleton { 11 , p , {{j} 1 j E 1I}).

115

The next lemma implies that the root sets, introduced in [E 2], Defini

tion 1.1, are generating sets with weight sequence 11.

1.6. Lemma + Let g c w (ID) be a Köthe set.

Then g is a generating set if f there exist a partition Q(1I) of ID and

sequences w E w+ (ID) and <: E !/,~ (1I), both with full support, such that

sup jEQ(m)

a • w -l (j) :;; À inf jEQ(m)

-1 b•w (j).

Proof:

•· Let (w,p,Q(lI) denote a skeleton of g. For each a E g, put

a = (. s ug a • W- l ( j)) ElI and put g = {'à j . a E g}. From Coro llary JEQ(m) m ,..,

1.4 it follows that w-1 • g ,.., g E> Q(IT).

Let <: E .11,~ (ll) denote a p-multiplier. Then <: is also a g-multiplier.

Hence there exist À > 0 and b E g such that

~ -1 -1 J.JI <: • a(m)xQ(m) :;; Àb •w •

Hence

V 7-1 (m) mEli : "

-1 -1 sup a•w (j) ~À inf b•w (j). jEQ(m) jEQ(m)

<=. Let a, a E g, and g be defined as in the previous part of this -1 ,..,

proof. Lemma 1. 3 and Corollary 1 • 4 imp ly that w • g ,.., g @ Q(JI). - -1 ,..., -1....... ....., rJ

Evidently, g ;s <: • g. The assumption implies tha.t <: g :;;, g. So g

is a moulding set and g is a generating set. o

We mention the following important properties of generating sets.

1. 7. Lemma

Let g c w +(ID) be a generating set with skeleton (W, p, Q(JI)) and let

v E w+(ID) have full support. Then

a. The set v • g is a generating set with skeleton ((v•w),p,Q(lI}).


b. Tbe set g# is a generating set with skeleton (w-1 ,p#.Q(JI)).

c. The set g is #-symmetrie iff the set p is #-symmetrie.

Proof:

-1 -1 a. The equality (v•w) • v • g = w • g implies that v • g is a generating

set with skeleton ((v•w),p,Q(Il)).

b. By Lemma 1.2.e. we have (w-1 •g)# ,..,p#@Q(JI). Since (w-1 •g)# =w•g#

and since by Lemma III.1.14 the set p# is moulding. the set g# is

generating with skeleton (w-1 ,p".Q(Il)).

c. Applying of b. twiee, we get w-1 • l# ...... p## E> Q(Il).

1 2 . l' . ## 'ff -l -l ## Lemma • i.mp 1es p ..., p 1 w • g ...... w • g • ## • ## Hence p ,.., p 1ff g "' g • c

An essential tool in the construction of the analytie function spaces

in this ehapter is the lifting operation up. which maps sequences on lD

onto sequenees on lM(JD) and generating eones in w +(ID) onto generating

sets in w+(:M(:ID)). We introduce this operation now.

Let F be an analytic function on tp(JD). Then the function 6aF: x 1+ F(a•x)

is also analytie on q>(JD) for all a E w+(JD). For all s E lM(lD) we have

For this reason, the sequenees (a5

) SElM(ID) with a € w +(ID) play an

important role.

1. 8. Definition

For each a E w+(ID) the sequence up[a] E i/(JM(ID)) is defined by

up[a](s) • a5 , s E lM(ID~. For each g c: </(ID) the set up[g] c: w+(:M(ID)) is defined by

up[g) = {up[a] 1 a E g}.

The up operation bas the following properties: Let a,b E w+(ID).

(1.3) VÀ>O Lup[A11](s) = Àlsl1, s E lM(ID).

(1.4) a S b "up[a] ~ up[b] •


( 1. 5) <3À>O : up[a] ;;; À up[b]) ,.. a ;;; b •

(1. 6) up [a•b] = up[a] • up[b]

(1. 7) up[aP] = (up[a])P , p E lN •

( 1. 8) -1 up [a ] (up[a]) -1

The up operation has a handicap also. In general, a ~ b does not imply

up[a] ~ up[b]. Cf. property (1.5). So fora quasi-directed set 9, in

general the set up[9] is not quasi-directed.

EXAMPLE: Consider the quasi-directed set 90 = {X E ~(lN) 1 lxl1

= 1}.

Let e. (o .. ).ElN' i = 1,2. Let À> 0 and c E w+(lN) such that 1. l.J J

up[ei];;; À up[c], i 1.,2. Property (1.5) implies that c(1);;; 1 and

c(2) ;;; 1. So lcl 1 ;;; 2 and c f 90. The set up[90 ] is not quasi-directed.

The set up[9] is directed if 9 is directed. We restrict ourselves to

sets 9, which are cones.

1. 9. Lemma

Let g c w +(ID) be a conic Köthe set.

Then up[9] is a Köthe set in w+(lM(ID)).

Proof:

We prove that up[9] is 1. separating and 2. directed.

1. Lets E lM(ID). For each j E ID[s] there exists a. E 9 with a.(j) -1' O. J J

Since ID [s] is a finite set and 9 is :a cone, there is a E 9 such that

a. ;;; a for j E ID[s). Then J

up [a] (s) n [a(j) ]s(j) -1' o . jEID [S]

2. Let a,b E 9. Since g is a cone, there exists c E 9 with a ;;; c and

b;;; c. So up[a] ;;; up[c] and up[b] ;;; up[c]. a

We want to show that up maps generating cones in w+(ID) into generating

sets in w+(lM(ID)). For this purpose the next theorem is fundamental.


1. 1 O. Theorem +

Let p € lN and let a € R,P(ID) with lalp < 1.

Then up[a] E t+(M(ID)). p

Proof:

By property (1.7) it follows that we may restrict ourselves top= 1.

For each j E ID we have a (j) < 1. Let (ID ) ElN denote an exhaustion of . q q ID and let j denote the only element of ID '-ID

1• Then q q q-

!up[a] l 1

00

l as = 1 + l l as SEM(ID) q=1 SElM(ID) ,j EID(S]c:ID .

00 00

q q

00 q k l n [a(jt)) t =

k =1 !=1 q

00 q = 1 + J: a(jq) / n (1 - a(jt)) •

q=l !=1

Since lal 1 < 1, the sequence (nÎ=t (1 - a(jR,)))qElN is nonincr_easing and

bas limit 13 >O. Hence !up[aJ! 1 ~1+ 1a1,tf3. o

1.11. Lemma

Let p c: w+ (:0:.) be a moulding cone and let <:: E t7 (:0:.) be a p-multiplier.

Then up[p] is amoulding set in w+(lM(ID)) and there exists À> 0 such

that up[À<;:] is a up[p)-multiplier in i;(M(ID)),

Proof:

Lemma 1.9 yields that up[p] is a Köthe set. So, we only have to prove

that up[p] bas a up[p]-multiplier in l~(lM(ID)). Let À> 0 such that

IÀC:::l 1 ~ 1. Si~ce pis a cone, À<:: is a p-multiplier. Further, Theorem

1.10 implies that up[À<;:j E t7(:M(ID)), We have p f':J (À<;:)•p, i.e.

p ~ (À<;:)•p and p ~ (Àè;)•p. So, up[p] "'up[(ÀC:::)•p) • up[À<;:] • up[p],

Hence up[À<;:] is a up(p]-multiplier in t7(lM(ID)). o

1. 12. Theorem + Let g c:: w (ID) denote a generating cone with skeleton (w,p,Q(:O:.)).

We introduce a partition R(:M(ID)) of :M(ID) as follows. For each

t E lM(][) let


Then up[g] is a generating set with skeleton (up[w],up[p],R(JM(lI))).

Proof: Without loss of generality we may assume that p is a cone. Hence

119

-1 . -1 w • g FI:$ p © Q(lI) and by property {1.4) up(W • g] - up[p 8 Q(Il)] or

equivalently (up[w])- 1 • up[g] - up[p © Q(Il)].

We prove that up[p 8 Q(lI)] = up[p] © R(JM(lI)). Let c E pand let

s E JM(ID). Let t 0 E JM(Il) fixed by t 0 (m) 1s•XQ(m)1 1 , m E n:. Then

up [mE}'.II c (m)xQ (m)] ( s) = mnElI n [ c (m)] s (j) = jEID[s]nQ(m)

n [c(m)]to(m) = up[c](t0

) mEil[t0J

= I up[c](t)xR(t)(s) • tEJM(E)

So, for all c E p the following equality holds:

up( I c(m)xQ( )] = I up[c](t)xR(t) mElI m tEJM(1I)

We have found that (up[W])- 1 • up[g] ""up[p] © R(JM(II)).

Because of Lemma 1.11, up[p] is a moulding set, whence up[g] is a

generating set with skeleton (up[w] ,up[p],R(JM(lI))). o

Finally, we say something about #-symmetry and the up operation.

In Chapter III, Section 1, the positive sequence sets have been classi

fied into three types. The up operation does not disturb this classifi

cation too much.

1. 13. Lemma + Let g c w (ID) be a conic Köthe set. Then

a, The set g is type I or type II iff the set up[9] is type II.

b. The set g is type III iff the set up[g] is type III.


Proof:

a. Let g be type I or type II. Then there exists a sequence (cn)n€lN in

g with en ~ cn+l and with the property that for each a € g there

exists n € lN with a ~ en.

For each n € lN put dn = up[cn]· Then up[g] ,...., {dn 1 n € JN}. We

prove that up[g] is not type I.

Assume up[g] ,.., {dn 1 1 ~ n ~ l} for some l € JN. Then there exists

µ > 0 such that up[2ct] ~ µ up[cil· For all j € ID and p€ lN we have

up[2ct](pej) ~ µ up[ct](pej) or equivalently zP[ct(j))P ~ µ[ct(j)]P.

So for all j € ID we find c t (j) " 0, which is absurd. Hence up [ g] is

type II.

Conversely, let up[g] be type II. Then we can choose a sequence

(cn)n€lN in g, such that up[g] "' {up[Cn] 1 n E lN}. Property (1.5)

implies that g "' {c 1 n E JN}. So g is type I or type II. n

b. This statement is a corollary to statement a.

1. 14. Corollary + Let g c: w (ID) be a type I or type II conic Köthe set.

Then the set up[g) is #-symmetrie.

Proof:

By Lemma 1.13 the set up[g] is type II. Lemma 1.9 yields that up[g] is

separating and directed. Because of Theorem III.1.8 the set up[g] is

c

#-symmetrie. c

EXAMPLE: The set IP+ (ID) is a type II, conic Köthe set. So up[lj)+ (ID)] is

a #-symmetrie Köthe set.

§ 2. Sequence spaces

Tbis section deals with inductive limits G. d[g] and projective limits i.n

G r .[g], subspaces of w(ID), induced by a Köthe set gor a generating p~ .

set g c w+(ID). The inductive limit and the projective limit of function

spaces, find{g] and fproj[g], which will be introduced in Section 5, can be considered as (ray-) analytic function spaces on these G .[g] and prOJ Gind[g], re~pectively. For generating sets g it will appear that Find[g]


and f .[gJ inherit topological properties of G .[g] and G. d[g]. proJ proJ in respectively.

This section is devoted to the construction of G. d[g] and G .[g] in proJ and some topological properties of these spaces. The spaces G. d[g]

i.n .

121

and G .[g] are sequence spaces. So we give some relations between our prOJ theory of inductive and projective limits and the theory of sequence

spaces.

First, we introduce the notion of sequence spaces.

2. 1. Definition

A subspace Vof w(ID), which is endowed with a L.C. topology, is a

sequence space if for each j E ID the evaluation v 1+ v (j), V E V. is

continuous.

We shall introduce sequence spaces G. d[g] and G .[g] by means of the in pro] construction of the inductive limits and projective limits of Section

111.2. These sequence spaces G. d[g] and G .[g] can also be described l.n prOJ

as inductive limits and projective limits of subspaces in w(ID), which

will be presented now,

2. 2. Definition

Letafw+(ID).

The Hilbert space .Q,2 [a;ID] is defined to be the space a • .Q,2 (ID), endowed

with the inner product

(x.y) .Q,2[a;ID] -1 -1 (a ·x. a •y) 2

The semi-inner product space i;[a;ID] is defined to be the space

a -l • .R.2 (ID) + ©a • w(ID), endowed with the semi-inner product

2. 3. Definition + Let g c w (ID) be a Köthe set.

-1 x,y E a • t 2(ID) + •a • w(ID) •

The inductive limit G. d[g] and the projective limit G .[g] are in proJ defined by


and

Cf. Definitions III.2.8 and III.2.t9.

We mention the following properties of these limits:

2. 14. Lemma + Let g,h c: w (ID) be Köthe sets with g ""h. Then

G. d[g] = G. d[h] 1n i.n

and

Prooi:

Apply Lemmas III.2.9 and III.2.20.

2.5. Lemma + Let g c w (ID) be a Köthe set. Then

and

Proof:

lim ind Jl.2

[a;ID] a e: g

+ G • [g] = lim proj Jl.

2[a;ID] •

pro1 a e: 9

Let b € w+(ID). The space H[b] used in Definition III.2.6 equals

0

Jl.2[b;ID]. The space H+[b] used in Definition III.2. t7 equals Jl.;[b;ID]. a

(2.1) The spaces G. d[gJ and G .[g] with the pairing in proJ

<x,y> == l. x(j) y(j) • j€ID

(x,y) € G. d[g] x G .[g] i.n proJ

constitute a Hilbertian dual system. Cf. Definition III.2.27.


2. 6. Lemma + Let g c w (ID) be a Köthe set.

The spaces G. d[g] and G .[g] are sequence spaces. 1n proJ

Proof:

We restrict ourselves to. the first case. Let j E. ID. Since e. E. G . [g], J prOJ

we conclude that x 1+ x(j) = <x,ej>' x E Gind[g], is continuous because

of Theorem III.2.29.

Next, we study the spaces G. d[g] and G .[g] where g c:w+(ID) is a 1n pro3

generating set. All results are based on the following theorem.

2 • 7. Theorem + Let g c w (ID) be a generating set with skeleton (W,p,Q(:U:)). Then

and

Proof:

By assumption, g "'W • (p 0 Q(n:)). Lemma 2.4 implies that

and

We use the notations from Chapter II!, S.ec tion 2. For m € n:, let

Hm = i 2 [w·xQ(m)'ID] and then H = Q,2 [w;ID].

Fix c E. w+(l[). The space H[c] introduced in Definition III.2.6 equals

i 2(lxn€n C(m)W • XQ(m) ;ID] and the space H+(c], introduced in Definition

III.2.17 equals i;c~][ c(m)w • XQ(m>'ID].

By Lemma 2.5 we have

Gind(g] lim ind li(c] l[p; (R,2 [w•xQ(m); ID ])m€1L] c E p

and +-

Gproj[g] lim proj 11 [c] = P[p; U 2[w·xQ(m) ;ID])mE.n:l C E p

0

a


We list some results on these inductive/projective limits. The proofs

are omitted, because with the preparations of Section 1 they are trans

lations of results on the inductive/projective limits introduced in

Section III.2.

(2.2) + Let g c w (ID) be a Köthe set.

a. The space Gind[g] is barreled and bornological (Theorem III.2.10).

b. The space Gproj[g] is complete (Theorem III.2.21).

(2.3) Let g c: w+(ID) be a generating set.

a. The inductive limit topology of G. d[g] equals the topology generated # . 1n

by the seminorms Pa• a Eg , defined by pa(x) = la•xl 2, x € Gind[g].

(Corollary III.2.13.)

b. The space Gind[g] is nuclear iff Va€g VbEg#: a•b E t 1(ID). (Theorem III.2.16.)

c. A bounded (compact) subset W of G • [g] is of the form a • .B where a . prOJ belongs to g# and B is a bounded (compact) subset of t

2(ID).

(Theorem III.2.23 and Corollary III.2.24.)

d. Gproj[g] is semi-Montel iff VaEg Vb€g# : a•b E c0 (JD).

(Theorem III.2.26.)

e. The space Gind[g] is a representation of the strong dual of Gproj[g].

(Theorem III.2.30.)

(2.4) + Let g c: w (lD) be a #-symmetrie generating set.

# # a. Gind[g] = Gproj[g ], Gproj[g] = Gind[g. ],

The spaces G. d[g] and G .[g] are reflexive. · 1n - proJ (Theorem III.3.1.)

b. The space G .[g] is a representation of the strong dual of G1.nd[g]. proJ (Theorem III.2 .• 31.)

c. If VaEg VbEg# : a•b € c0 (ID), then both spaces Gind[g]and Gptoj[g] are Montel.

(Theorem III.3.3.)


d. If VaEg VbEg# : a•b E .R-1 (ID), then both spaces Gind [g] and Gproj [g]

are nuc lear.

(Theorem III.3.4.)

Lemma 2.6 states that G. d[g] and G .[g] are sequence spaces. We . in proJ give some relevant topics of the sequence space theory as founded by

KÖthe. These topics can also be found .in Kamthan and Gupta, [KG],

Chapter II.

Let S c w(ID) be a sequence space.

* The space S is normal if for each u E S the normal cone

{V E w(ID) 1 IVI ~ lul} of u is contained in S. It turns out that

the space S is normal iff .R,00

(ID) • S = S.

* The space Sx c w(ID) is defined by

Sx = {v E w(ID) 1 VUES u•v E i1

(ID)} •

The space Sx has been put forward as a natural dual of S. It is

cal led the Köthe dual or the arass of S. We have S c Sxx and

Sx = Sxxx. The space Sx is normal.

* The space Sis a Köthe spaae if S = Sxx. So Sx is a Köthe space.

* For each v E Sx the seminorm !tv on S is defined by !t/U) = lu~!v 11

,

U ES. The seminorms !tv' V E Sx, generate the normal or Köthe topo

logy of S.

In the following lemma we connect facts on Köthe duals with our results

on inductive/projective limits. • x +-

First of all: (.R-2

[a;ID]) = .R-2

[a;ID].

2.8. Lemma + Let g c w (ID) be a generating set. Then as sets

a.

b.

# = G. d [g ] in:

G [ #] = G. [g##] proj g ind

126

c.

Proof:

G. [g##] ind


-1 Let g have a skeleton (w,p.Q(lI)). Then w • g is a root set, i.e. a

generating set with weight sequence n. Further we have

and

-1 =w•G.d[W •g] in

-1 -1 Gproj[g] = w • Gproj[W • g] •

The remaining part of the proof consists of an application of [E 2], -1 -r Proposition 8.12, to G. d[w • g] and G • [w • g]. o in proJ

So the space G .[g] is always a Köthe space. The space G;nd[g] is a prOJ "

Köthe space iff the set g is #-symmetrie. This shows again the relevance

of the #-symmetry condition.

2.9. Lemma + Let g c: w (lD) be a generating set with skeleton (w,p,Q(lI)). Let

è; E R-1 (:JI.) denote a p-multiplier, such that ~Il ~(m)XQ(m) E R-1 (D).

(This condition implies that the spaces G. d[g] and G .[g] are in proJ nuclear.) Then

a. The inducti.ve limit topology of G. d[g] is equal to the normal topol.n

b. The projective limit topology of G .[g] is equal to the normal topo-prOJ logy of G .[g]. prOJ

Proof:

Just like in Lemma 2.8 we must replace the generating set g by the root -1 set W • g. Further, see [E 2], Lemma 8.16 and Corollary 8.17. o

The Lemmas 2.8 and 2.9 show that there is an overlap between our theory

of inductive/projective limits and the Köthe space theory.


§ 3. A sequence space representation of A(q>(ID)) and A(w(ID))

In this section we introduce an injective mapping Seq from A(q>(ID)) into

w(JM(ID)). After some preparation in Section 4. in Section 5 we introduce

inductive limits Find[g] and projective limits Fproj[g] of analytic

functions. which belong to A(qi(ID)). So, having defined the mapping Seq

in A(<.p(ID)), the mapping Seq yields sequence space representations of

F. d[g] and F .[g]. These representations deepen our insights in the in proJ topological structure of the spaces f. d[g] and F .[g]. in proJ

This section is divided into two parts. The topics in the first part

are the sequence space representation of A(q>(ID)) and A(w(ID)) induced

by the mapping Seq. The L.C. spaces q>(ID) and w(ID) are nuclear. In

literature one can find extensive discussions on spa.ces consisting of

analytic functions on nuclear L.C. spaces. Cf. [Di 1] and the references

therein. We present a sequence space representation of A(q>(ID)) and of

A(w(ID)) as a demonstration of our techniques. The second part of this

section is devoted to the strong duals of A(q>(ID)) and A(w(ID)) and is

not connected with the remaining part of this chapter.

We introduce the mapping Seq. To this end we need the differential

operator;; 35 , s E :M(ID), defined on A (qi(ID)). Cf. (4. 6) in Section I. 4.

3. 1 • Definition For each F E A(<t>(ID)) the s.equence Seq[F] E w(lM(ID)) is defined by

Seq[F](S) "-1- [as F] (lfl) • Tsï

S € lM(ID) •

We give some properties of the operator Seq, in which the following

functions occur:

- The functions <lis, s € :M(ID), on<.p(ID) with cl>5

(x) --1-x5•

rsr - The function K on <.p(ID) x <.p(ID) with K(x,y) • exp(x,y) 2•

(3.1) Each F € A(q>(ID)) satisfies

v E (ID): F(X) = i Seq[F](S)<lis(X) X !.P SE:M(ID)



sets in <P(lD). Cf. Theorem I.4.19.

(3.2) LetFEA(q>(ID)) and let a€w+(ID).

Then the function 0a F with [0a F](x) = F(a•x) belongs to A((jl(ID)) and

Seq [0a F] = up[a] • Seq[F].

(3.3) The spac.e q>(lD) is dense in 4!R.2 , whence F i+ Flq>(ID)' F E S(.R.2 (ID))

is a unitary operator from S(t2(lD)) onto H(1P(lD),K). Cf. Theorem

II.1.14. The results on S(t2(1D)) in Section II.2 imply that

H(1&>(ID),K) c: A(cp(ID)) and that H5

1 s E lM(ID)} is an orthonormal basis

in H(q>(ID) ,K). So for F € H(qi(ID) ,K} and s E lM(lD) we have

Seq[FJ(s) = (F ,'1\)H(\&)(ID) ,K) •

Property (3.3) leads to:

3.2. Lemma

The operator Seq is a unitary operator from H(q>(ID) ,K) on to t2

(lM(ID)).

Proof:

The proof is based on the relation Seq[F](S)

Next, we introduce the ~nnounced sequence space representations of

A(q>(ID)} and A(w(ID)). It turns outt that the ranges of A\(1P(lD)) and

A(w(ID)) under the operator Seq are as sets equal to projective limits

of the type G .[9] with ga generating set of the special type intro-prOJ

duced now.

3. 3. Definition

Let a E w+(ID) and g c w""(m).

The sequence v E w(lM(ID)) is defined by

V (S ) " 1 / ISf .

The sequence ev[a] E w+(lM(lD)) and the sequence set ev[g] c w+(lM(ID))

are defined by

ev[a] = v • up(a] and ev[g] = v • up[g] •

c


REMARK: + For all a ,b E. w (ID) we have up [a] • ev [b]

3.ll. Lemma + Let g c w (ID) be a generating cone.

ev[a•b]

Then the set ev[g] is a generating set in w+(lM(ID)).

Proof:

Theorem 1.12 implies that up[g] is a generating set in w+(ll:(ID)).

Since V = (1//ST)SElM(ID) has full support, we get by Lemma 1.7 that

129

ev [g] v • up [g] is a generating set. a

So, ev[cp+(ID)] and ev[w+(ID)] are generating sets.

Before we state the theorem on the sequence space representations of

A(cp(ID)) and A(w(ID)), induced by Seq, we recapitulate some relevant

results from Section I.4.

(3. 4) A complex valued function F on cp(ID) is analytic on q>(ID) iff F

is ray-analytic on cp(ID).

(3.5) A co!ltplex valued function G on w(ID) is analytic on w(ID) iff G

is ray-analytic on w(ID) and there exists a finite subset 1F c ID such

that for all x E w(ID) G(x) = G(XlF • x).

(3.6) The mapping Go+ Glcp(ID) is an injection from A(w(ID)) into

A(cp(ID)). We identify the function G E A(w(ID)) with its restriction to

cp(ID). So A(w(ID)) c A(cp(ID)).

We come to the main result of this section.

3. 5. Theorem

a. The operator Seq maps A(cp(ID)) bijectively onto G • [ev [q>+ (ID)]]. prOJ

b. The operator Seq maps A(w(ID)) bijectively onto G .[ev[w+(ID)]]. prOJ

Proof:

a. Fix F E A(cp(ID)). For all X E q>(ID) we have

F(X) = l Seq[F](S) • 4> 5 (-X) = l (Seq[F] • ev[X]) (S) • sElM(ID) SElM(ID)


From (3.1) it follows that Seq[F] • ev[a] E R.2 (lM(ID)) fo;r each

a E q:,+(ID), whence Seq[F] EG .[ev[q>+(ID)]]. proJ +

Conversely, let f EG .[ev[q>+(ID)]]. Let C E R.1

(ID) denote a 1.fJ+(ID)-proJ +

multiplier with ICl1 < 1. Then up[?;] E R.1 (lf(ID)) by Theorem 1.10.

Let a E q>+ (ID). Since C bas full support, we have

-1 f • ev[a] = up[CJ • f • ev[( • a] •

-1 + -1 Because ( • a E <P (ID), the sequence f • ev[C • a] belongs to

R.00

(:M(ID)), whence f • ev[a) E ~(M(ID)).

The series lsE:JM(lD) (f • ev[x]) (S) is absolutely and unifonnly con

vergent in x on compact sets in q.i(ID), so the complex valued function

F on q>(ID), defined by F(x) = lsEM(ID) (f•ev[x])(s), x E ql(ID). is

analytic on all finite dimensional subspaces W of q>(JD). Corollary

I. 4. 16 implies that F E A(l,fJ(ID)). We have Seq [F) = f.

Evidently, the operator Seq is injective.

b. Let GE A(w(ID)). There exists a finite subset IF of ID such that for

all x E w(lD), G(x) = G(xIF• x). This leads to the relation

Seq[G) = Seq[G]. up[xIFJ •

Since fora E w+(ID) the sequence XIF• a belongs to q>+(ID), we have

Seq[G] • up[a] • Seq[G] • up[xIF• a] € R.2

(:M(ID))

+ So, Seq [G] E G . [ev [ (ID}]]. By state-proJ proJ

ment a. of this theorem there exists G € Á(q>(ID)) with Seq[G] = g. We prove that G belongs to A(w(JD)) by proving that there exists a

finite subset IF of ID, such that for all x E w(ID) G(X) = G(XIF• X).

In terms of g = Seq[G] this means that g = g • up[xIF].

So suppose the opposite, i.e. for every finite subset IF of ID we

have g # g • up[XIF).

First note that for all subsets IF of ID and all s E M(JD)

= \ 1 ,

( 0 •

ID[s] c:: IF

ID[S] </:. IF


Since g # Gl, there exists s 1 E 1M(ID) with g (s 1

) # 0. For J/, > 1 we

proceed inductively. We define SJ/, E :M(ID) as fellows;

Since IDJ/,-l = u~:: ID[sk] is a finite subset of ID, there exists

SJ/, E 1M(ID) with the properties

Next we define a sequence b E w+·(ID) and we will prove that

g• èv[b] f. J/,00

(1M(ID)). Thus we arrive at a contradiction. Let (jJ/,)J/,EJN

denote a sequence in ID with jJ/, E ID[SJ/,] and jJ/, '/. IDJ/,-l' The jJ/,'s are

all distinct.

The sequence b E w+(ID) is defined by

b (j) = 1 ' j # j J/, for all J/, E 1N

J/, E 1N •

For all J/, E 1N we have 1 (g • ev[b]) (SJ/,) 1 ~ !/,,

The previous theorem suggests a way to define L.C. topologies in

A((j)(ID)) and in A(w(ID)). In the sequel we consider A(\P(ID)) and

A(w(ID)) as L.C. spaces with the coarsest topology, which makes Seq a

continuous bijection from A(\P(ID)) on to G . [ev [\P +(ID)]] and from + prOJ

A(w(ID)) onto G .[ev[w (ID)]], respectively. We remark that both topo-proJ

logies are finer than the compact open topology, i.e. the topology of

uniform convergence on the compact subsets of (j)(ID) and w(ID), respec

tively.

The final part of this section deals with the strong duals of A((j)(ID))

and A(w(ID)). +

As we have seen, the sequence space G. d[ev[(j) (ID)]] can be regarded as + in

the strong dual of G .[ev[(j) (ID)]]. Cf. Result (2.3). So the natura! proJ

candidate for a representation of the strong dual of A((j)(ID)) is given

by Seq-1 [G. d[ev[(j)+(ID)]]]. Note that G. d[ev[(j)+(ID)]] is contained in in in

G • [ev[w+ (ID)]]. Hence, by means of the mapping Seq, we obtain a prOJ

description of the strong dual of A((j)(ID)) in terms of a subspace of

a

132 IV Spaces of analytic f unctions

A(w(ID)). Similarly, Seq- 1[Gind[ev[w+(ID)]]], a subspace of A(tp(ID)),

is a description of the strong dual of A(w(ID)).

Our results on the strong duals of A(q>(ID)) and A(w(ID)) are natural

infinitely dimensional analogues of a result by J.P. Antoine and

M. Vause [AV]. In [AV] they d iscuss a pairing between two analytic

spaces on t. They deal with the space A(t) consisting of all analytic

functions on t and with the space Exp(t), consisting of all entire

functions F of exponential type, i.e.

See also example 6 in [EGK].

Here we find ourselves in similar. situations. We remarked already that

the strong dual of A(q>(ID)) can be represented by a subspace of A(w(ID)).

3.6. Theorem

Let F E A(w(ID)).

Then Seq[F] EG. d[ev[qt(ID)]] iff in

Proof:

Assume Seq[F] € G. d[ev[q>+(ID)]]. Then there exist a E Q>+(ID) and l.ll

g € t 2 (:M(ID)) such that Seq[F] = ev[a] ··g and for all X E w(ID) we have

IF(x) 1 = 1 I +,- g(s) (a•x) 5 1 ~ sElM(ID) .

~19100 • I ~(a•lxl) 5 = S€1M(ID) S'

lgl00

• exp(<a, lxf>)

Conversely, let a € Q>+ (ID) and y > 0 such that

VxEw(ID) : IF(x) 1 ~ y exp(<a, lxl>) •

. + We show that Seq[F] = ev[b] • g for some b E Q> (ID) and g E i

2C:M(ID)),

With the aid of the Cauchy integral (cf. Lemma I.4.18) we obtain for


all r € w +(ID) with ID[r] • ID and all s E JM(ID} the estimate

(3.7) 1 [ël 5 F](t))I ::> ysl exp(<a,r>)r-s •

Let S E JM(ID).

If there exists j E ID[S] with j </. ID[a] we substitute r = (),-1)e. + n

in equality (3. 7) and we obtain 1 [as F](ll} 1 ::! y s! À-s(j). Taking ÀJ + 00,

it follows that [3 5 F](~} = O.

If ID[S] c: ID[a], we substitute r = s·a-1 +os in inequality (3.7) and

we obtain llosF](ll)I ::! ysl exp(lsl1)s-s as ::!y(ea)5

• So !Seq[FJI ::!>

::! y• ev(ea].

Putµ = 2 • #(ID[a]). Then l 11/vl 2 < 1 and up[11/µ] E J1,2 (:M(ID)).

Put b = µea and g = Seq[F] • ev[b-1], Then b E q>+(ID) and since

lgl ::! Y up[11/lll, g E t2

(JM(ID)). Further we have Seq[F] = ev[b] • g. c

The strong dual of A(w(ID)) can be represented by a subspace of A(q>(ID)).

We have a similar result as in Theorem 3.6:

3. 7. Theorem

Let F E A(q>(ID)).

Then Seq[F] E Gind[ev[w+(ID)]] iff

3bEw+(ID)3y>O '°"xEc.p(ID): IF(X)I ::! y exp(<b,lxl>).

Proof:

The proof is based on similar argtullents as the proof of Theorem 3.6. a

The two previous theorema lead to the introduction of the spaces of

functions of exponential type on w(ID) and c.p(ID), respectively.

3. 8. Definition

The spaces Exp(w(ID)) and Exp(c.p(ID)} are defined by

and

Exp(w(ID)) = Seq-1 [G. d[ev[c.p+(ID)]]] 1n

-1 + Exp(tp(JD)) = Seq [G . [ev[w (ID)]]].

prOJ


The pairing between A(q>(ID)) and Exp(w(ID)) is given by

<F,G> F E A(q>(ID)) • G € Exp(w(ID)) •

The pairing between A(w(ID)) and Exp(q>(ID)) is completely similar.

§ 4. Elementary spaces in A(cp( ID))

+ In Section 2 we have proved that for a Köthe set g c w (ID)

G. d[g] = lim ind fl2[a;JD] and G • [g] = lim proj fl+2[a;ID] • in a € g proJ a E g

This description of the spaces G. d[g] and G .[g] is the guide for J.n prOJ

the construction of our analytic function spaces F. d[g] and F .[g], in pro3 the compound spaces. For each a E w+(ID) we introduce two elementary

spaces, the Hilbert space F. d[a] and the semi-inner product space in f .[a]. These elementary spaces are the building blocks for the compro1 pound spaces F. d(g] and F . [g], viz.

J.n prOJ

F. d[g] = lim ind find[a] rn a E g

and F .[g] = lim proj F .[a]. proJ a E g proJ

This section consists of two similar parts, The first part deals with + the elementary spaces F. d[a], a € w (ID). Amongst others, we study

l.Il

continuous.functions on fl;[a;ID] which are extensions of the functions

in F. d[a] and we study the sequence space representation of F. d[a] in ].Il

induced by the operator Seq. The second part deals with the elementary

spaces F .[a]. Here we discuss similar topics as in the first part. prOJ

The elementary spaces F ind [a J

The spaces F. d[a] are functional Hilbert spaces determined by a special in class of functions of positive type on q>(ID).

ll. 1. Definition

Let a E w+(ID).

The function Ka E PT(q>(ID)) is defined by


x,y € q>(ID) •

The elementary space Find[a] is defined as the functional Hlilbert space

H(q>(ID) ,K3).

We denote K11

by K and F. d[11] by F. in

4.2. Lemma

Let a € w+(ID) and let FE Find[a].

The function 9 1

F belongs to F and a-

lt9 _1 FUF ~ UFHF. [a] • a ind

Proof:

Let V(q>(ID),Ka) denote the linear span <{Ic; 1 xEq>(ID)}>. a ~.Q, a

Let GE V(q>(ID),K ). Then G = lj=l a'j KYj· So,

and

0 _1

G

a

.Q,

L a. Ka•y. j=1 J J

Since V(<.P(ID),K3

) is dense in f. d[a], we find that 0 -l maps F. d[a] in a in isomorphically into F.

4. 3. Corollary +

Let a E w (ID) and F E F. d[a]. in The function Fis analytic on q>(ID).

Proof:

We only have to prove that F is ray-analytic on q>(ID). By Lemma 4.2 the

function 9 -1 F belongs to F. Lemma II.1.14 implies that there exists a a

function G € S(.Q,2(ID)) = H(R-2 (ID),exp( , ) 2 ) such that 8a_1 F = Glq>(ID)'

Since G is analytic on t 2 (ID), the function 9a-1 F is ray-analytic on

0

q>(ID). Hence F itself is ray-analytic on q>(ID). o


The following results justify the use of the elementary spaces Find[a]

as building blocks for inductive limits. Cf. Definition I.2.1.

4. 4. Theorem + Let a,b E w (ID).

Then a :;; b iff F. d[a] c::;.. F. d [b]. in in·

Proof:

Lemma II.1.10 states that

Assume Ka :;; µKb for some µ > 0. Then for all À > 0 and all j E ID

Hence a :;; b.

Conversely, let a :;; b. Since for all u E q>(ID), (a•u,a•u) 2 Lemma II.1.17 implies that Ka:;; Kb.

4. 5. Corollary + Let g c w (ID) denote a conic KÖthe set.

Then {F. d[a] 1 a E g} is an inductive system. in

Proof:

Since g is a cone, the set g is a directed set. For a,b E g with a :;; b,

0

Theorem 4.2 implies F. d[a] c-...F. d[b]. c in in

Each function F E F. d[a] can be extended uniquely to a continuous + in

function on R.2

[a;ID].

4.6. Theorem

Let aE w+(ID) and let F E F. d[a]. in . +

There exists precisely one continuous function G on i 2 [a;ID], such that

F = Glq>(ID)" The function G is ray-analytic on ,Q,;[a;ID].

IV Spaces of artalytic functions

Proof: ·

First we prove that F = 811 a F ~

Let x € cp(ID). Since F is ray-analytic, the func tion f, defined by

f(À) = F(11 3 • x +À Gla • x), À E q:, is entire. Since F € Find [a], there

exists y > 0 such that

Liouville's theorem implies F(x) = f(1) = f(O) = F(11 • x). Hence a F = 8tl F.

a

137

The function Sa-t F belongs to F. There exists H € S(.e,2 (ID)) such that

8 -1 F =Hl (ID)• Cf. the proof of Corollary 4.3. a qi + . +

We define G on t2

[a;ID] by G(x) = H(a•x), x E R.2 [a;ID].

The function G is continuous on .e.;[a;ID] and for all x E qi(ID)

G(x) = H(a•x) = F(na • x) = F(x) •

Since qi(ID) is dense in R.;[a;ID], there exists exactly one such function

G. Remark that G is ray-analytic on .e.;[a;ID]. o

Since the space F. d[a] is contained in A(IQ(ID)), the operator Seq in

induces a sequence space representation of Find[a].

Il. 7. Theorem

Let a E w+(ID).

The operator Seq is an isometrie isomorphism from F. d[a] onto lil

t2

[up[a];:M(ID)].

Proof:

Let F E F. d[a]. Then l.Il

Seq[F] = up[a] • Seq[S _1

F] € R.2

[up(a];:M(ID)] • a

-1 -1 Conversely, let f E R.2 [up[a];:M(ID)]. Put F =Sa (Seq [up[a ] • f]).

-1 Then F E F. d[a] and Seq[F] = up[a] • up[a ] • f = f. in By Lemmas 4.2 and 3.2 we have for all F € F. d[a] in


USeq [F] Il R.2

[up [a] ;1M(1P)] 11Seq[9 _1 F]llR, (1M(ID)) = a 2

119a-1 FllF = llFllF. [a] • ind

The elementary spaces F . [a] prOJ

The spaces F .[a] are semi-inner product spaces defined in the followprOJ

ing way.

4. 8. Definition

Let a € w+(ID).

The elementary space F .[a] is defined as the space prOJ

{F € A(11>(1P)) 1 ea F € F}

endowed with the semi-inner product

Sometimes elementary spaces of bot.h types coincide.

4.9. Lemma + Let a,b E w (ID).

Then a • b = 11 iff Fproj [a] Find [b].

Proof:

Since for all x € q>(ID) and F € F . [a] prOJ

the subspace 911

(F . [a]) of F . [a] is a functional Hilbet't space a prOJ -1 prOJ witb reproducing kemel Ka We have

and

iff a • b = 11 •

0

IJ


In particular, Fproj[n] = Find[n] =F.

The next theorem is the analogue of Theorem 4.4.

4.10. Theorem + Let a,b E w (ID).

Then a:;; b iff Fproj[b] c;..fproj[a].

Proof:

""'·Let F € F .[b]. Then 0bF E f. Since b-1 ·a:;; n, the operator proJ 0 -l 1 F is a bounded diagonal operator on F. So 0 F = 0 _1 (0b F) E f b ·a a b •a

and

". Fix j € ID [a] , For r > 0 we define Gr € A(q>(ID)) by Gr (z) = = exp(h• z2(j)), z E qi(ID). Lemma II.3.10 implies that G € f iff

r 0 < r < 1. So 0a Gr € f iff 0 < r < a-2

(j). Hence 0b Gr f. f for

r ~ a-2(j). We find that b(j) r O and a-2(j) ~ b-2(j). Soa :;; b. a

4. 11. Corollary + Let g c w (ID) denote a conic Köthe set.

Then {f .[a] 1 a Eg} is a projective· system. prOJ

Proof:

Since g is a cone, the set g is directed. For a,b E g with a :;; b,

Theorem4.10impliesF .[b]c .... F .[a]. a proJ proJ

By Theorem 4.6, to each function F € F. d[a] there is associated pre-1n

cisely one continuous function on ~;[a;ID]. To each function F E F .[a] there is associated precisely one analytic prOJ function on ~2 [a;ID].

4. 12. Theorem

Let a E w+(ID) and let F E F .[a]. prOJ There exists precisely one function G E S(t2[a;ID]) such that


Proof: -1 -1

The function Flna•q:>(ID) belongs to H(n8

•q:i(ID);exp[(a •.,a •.)2]).

By Theorem II.1.14 there exists G E s(R,2 [a;ID]) such that

Fin •q:>(ID) =Gin •<tJ(lD}. a a

The function G is analytic on t 2 [a;ID] and since na• q:>(ID) is dense in

t 2 [a;ID], the function G is uniquely determined. c

Finally, we deal with the sequence space representation of the elementary

space F .(a], induced by the operator Seq. prOJ

4.13. Theorem

Let a € w+(ID}.

..... a. The i>perator Seq -maps F . (a] continuously into t

2[up[a];JM(ID)].

proJ

b. The range Seq[f .[a)] equals proJ

• . + J!.2

[up [a] ;JM(ID)] n G • [ev [!.P (ID}]] • prOJ

Proof:

a. Let F € F • [a]. Then proJ

-1 (up[a]) • Seq [9a F] E

-1 + (up[a]) • J!.

2(JM(ID)) c .R-

2[up[a];JM(ID)]

and

= ISeq [98 F]DJ!, (JM(ID)) = 119a FllF = DFRF. [a] • 2 Llld

b. c:. Statement a. and Theorem 3.5 imply this inclusion.

=>. Let f € R-+2[up[a];JM(ID)] n G .{ev[q:i+(ID)]]. Put g = f - up[tla] •f. prOJ

By Theorem 3.5 there exist F,G € A(q:i(ID}) such that -1 -1

F = Seq [up[11a]· f] and G = Seq [g].


Then F + G E A(qi(ID)) and 0a (F + G)

and Seq[F +G] = f.

0aFEF.SoF+GEF .[a] · prOJ

§ 5. Compound spaces in A(qi(ID))

In this section we define inductive limits F. d[g] in F .[g] of elementary spaces for conic Köthe sets prOJ

and projective limits + 9 c w (ID). We call

a

these limits F. d[g] and F .[g] càmpound spaces. The elements of these · in pro3 compound spaces are analytic functions on qi(ID).

We mention two important properties of these new spaces:

First, the spaces Seq[F. d[g]] and Seq[F .[g]J are inductive limits in prOJ and projective limits, respectively, of sequence spaces of the kind

introduced in Chapter III. So results of Chapter III apply.

Second, the elements of F. d[g] and F .[g], being analytic functions in proJ on tp(ID), can be extended to analytic functions on G .[g] and to ray

prOJ analytic functions on G. d[g], respectively. in

The subdivision of this section is as follows

The first two parts deal with the compound spaces F. d[g) and the comin pound spaces F . (g], respectively, induced by conic Köthe sets prOJ g c u/ (ID). In the third part we study the compound spaces induced by

generating cones in w+ (ID).

We start with the inductive limits F. d[g]. l.n

5. 1. Definition + .Let g c w (ID) denote a conic Köthe set.

The compound space F. d[g] is defined by l.n

lim ind Find[a] • a E g

+ For each conic Köthe set g c w (ID) we have

+ + F. d[q> (ID)] c;... F. d[g] c;.. F. d[w (ID)] in in in


The space Find[g] is an algebra in the usual way.

5.2. Lemma

Let a,b E t/(ID), let F E F. d[a] and let GE f. d[b]. in in Then F • G E f. d[a+b] and in

IF •Gif. d[a+b] :; DFllF. [a]. llGllF. [b] in ind ind

Proof:

The proof is based on Lemma II.1.7. Put

a= 2 2 RFHF. [a] and 13 = llGllF. [b] •

ind ind

Let t E IN and a.. E 4:, x. E q>(ID), 1 ::> j ::> R.. Then . J J

J/,

y = 1 I a. F(x.}G(x.)1 2 ~ j=1 J J J

Now application of Remark (II.1.4) and Lemma II.1.20 gives

R,

y ::>. a.{3 I ëi"k a. j exp [ ( a • xk, a • x j ) 2 + ( b • xk, b • x j) 2 ] • k,j=1

Since (x,y} * exp[(a•x,b•y) 2 + (b•x,a•y) 2 ] is a function of positive

type, greater than (.i<,y) ~ 1, Lemma II. 1. 20 implies

Hence F • G E f. d[a+b]. By Corollary II.1.8 we get in

llF • Gllfi .. nd[a+b] ;:> llFllF. [a] • llGllF. [b] ind ind

c


5. 3. Corollary + Let g c w (ID) be a conic Köthe set.

a. For all F,G E F. d[g] the function F •G belongs to F. d[g). in in

b. For all GE F. d[g] the mapping F 1+ F • G is a continuous linear in operator from F. d[g] into F. d[g]. in in

Proof:

143

a. Let a,b E g. Since g is a conic Köthe set, there exists c E g, such

that a+b ~ c. So a product of two elements of Find[g] also is an element of F. d[g]. in .

b. Let G E F ind [b] for some b E g. Let MG denote the operator F i+ F • G.

Lemma 5.2 implies that the operato.r MG maps the space Find [a] con

tinuously into find[g]. Since MG is a linear operator, Theorem I.2.3

yields that MG is a continuous operator on Find[g]. o

We characterize the elements of F. d[g] in terms of a growth estimate. in

5. 4. Theorem + Let g c w (ID) be a conic Köthe set. Let F be a complex valued function

on <P(ID).

Then F belongs to F. d[g] iff there exists a € g and y > 0 such that in

Q,

1 ~ a. F"<X:TI z j=1 J J

Proof:

The statement is equivalent to:

The function F belongs to F. d[g] iff there exists a € g such that in F E F. d[a]. Cf. Theorem II.1.7. o in

The next theorem states that Find[g] can be identified with a subspace

of A(G • [g]). prOJ


5.5. Theorem + Let g c.: w (ID) be a conic Köthe set and let FE Find[g].

There exists a unique element GE A(G .[g]) such that Fis the prOJ restriction of G to qi(ID).

Proof:

There exists a E g such that F E F. d[a]. By Theorem 4.6 it follows 1.n

that F has a unique continuous ray-analytic extension H to !l;[a;ID].

Since G .[g] c;..!l+2 [a;ID], the function G = HIG [ ] is analytic on prOJ , g G .[g] and is uniquely detennined. proJ o

proJ

One of the main results in this section is the fact that the space

find[g] bas the same structure as the inductive limit Gind[up[g]] in

w(:M(ID)).

s. 6. Theorem + Let g c.: w (ID) be a conic KÖthe set.

The linear operator Seq maps Find[g] homeomorphicly onto Gind[up[g]].

Proof:

Lemma 4. 7 implies that the operator Seq maps F. d [a] homeomorphic ly in

onto !l2[up[a];lM(ID)]. Theorem I.2.3 implies that the operator Seq is a

homeomorphism from f. d·[g] onto in

lim ind !l2 [up[a];lM(ID)] = Gind[up[g]] • a Eg

If g is a conic Köthe set, then the set up[g] is a Köthe set, so the

topological properties of F. d[g] which are summarized at the end of in

this section, follow immediately from Section 2.

In the next part we consider the projective limits F .[g]. proJ

5. 7. Definition + Let g c.: w (ID) be a conic Köthe set.

The compound space F .[g] is defined by proJ

fproJ.[g] = lim proj F .[a] • a E g proJ

0


For each conic Köthe set g c w+ (ID) we have

We charactèrize the elements of F .(g] in terms of a growth estimate. proJ

s. 8. Theorem + Let g c: w (ID) be a conic Köthe set and let F denote a complex valued

function on cp(ID).

The function F belongs to F .[g] iff for alla€ g there exists y > 0 prOJ such that

v v R.EIN a.E4;,x.Ecp(ID), 1 J J

Proof:

". Since F € Fproj[g], we get that for alla€ g the function F belongs

to F . [a]. Hence, for all a E g, the function 0a F belongs to F. prOJ Apply Lemma II.1.7.

*'•Lemma II.1.7 implies that the function x 1+ F(a•x), x E cp(ID}, belongs

to F. If F is analytic on cp(ID), then F E F • [a] for all a E g. So prOJ

we only have to prove that F is analytic on cp(ID) or equivalently

that Fis ray-analytic on cp(ID). Cf. Theorem I.4~15.

Let u,v E cp(ID). Since g is a Köthe set, there exists a Eg such that

the supports ID[u] and ID[V] are contained in ID[a]. The function

À E C

is entire, 0

Just like the compound space F. d [g], the compound space F . [g] is an in proJ

algebra in the usual way.


5.9. Theorem + Let g c w (ID) be a conic KÖthe set. Let F,G € F . [g].

proJ Then the function F • G belongs to F • [g] and satisfies for all a € g

prOJ

Proof:

Let a € g. Put a = 110rzaFllF and 8 = 110v'2aGllF. We apply Lennna II.1.7.

Let R, € lN, aj € 4:, Xj € q>(ID), 1 ;;> j ;;> R,. Then

~ 2 1 I a.[ea F](x.) [0"'2 aG](xJ.//2)1 ;;;; j=l J . J Y.l.

Since 0/2a F € F, Remark (II.1.4) yields that

By Lemma II.1.20 we obtain that

So 0a (F • G) belongs to F and by Corollary II. 1. 8

Theorem 5.8 implies that the function F • G belongs to F • [g]. prOJ

Tbeorem 5.9 states tbat the product in F .[g] is jointly continuous, prOJ

in contrast with Corollary 5.3, which states tbat the product in F. d[g] in

is separately continuous.

c


Under the condition tbat g is type II, 'We will prove the analogue of

Theorem 5.5, which reads: Each F € F . [9] has a unique analytic prOJ

extension G on G. d[g]. For arbitrary conic power sets g we have the l.n

following weaker result:

5.1 O. Theorem + Let g c: w (ID) be a conic Köthe set. Let F € F . [g].

prOJ Then the function F bas a unique ray-analytic extension G on G. d[g],

l.n

147

such that for alla€ g the restriction Gl.e,2

[a;ID] belongs to sU2 [a;ID]).

Proof:

For alla Eg the function F belongs to F .[a]. So from Theorem 4.12 prOJ

it follows tbat for each a € g there exists a function Ga € S(R-2[a;ID])

such that Ga (x) = F(x) for all x E 11 a • ip(ID).

For all a,b E g we have that

because the functions Ga and Gb are continuous and their restrictions

tp 11 a • 11 b • ll)(ID) are equal (determined by F).

We define G on G. d[g] = u t 2 [a;ID] by J.n a€g

x E .e.2

[a; ID J , a € g •

Indeed, for alla Eg the restriction Gltz[a;ID] =Ga is analytic.

Since for each u,v E Gind[g] there exists a E g such that u,v E t 2 [a;ID],

the function À » G(u + ÀV) = Ga (u + ÀV) is entire. So the function G is

ray-analytic on Gind[g]. c

The fact tbat for each a € g the restriction GIR,2

[a;ID] is condnuous,

does not imply that Gis continuous on G. d[g], because the function G l.n

is not convex. Cf. Theorem I.2.3. However, under the additional condi-

tion that the conic Köthe set g is type II, we will prove that G is con

tinuous on Gind[g]. The proof relies heavily on the fact that the re

strictions GI Jl.z[a;ID] belong to S(Jl.2 [a;ID]). We give a proof in two

steps.


S. 11. Theorem + Let g c w (:ID) be a conic Köthe set, let C denote the closed unit ball

in R.2 (ID) and let G be an extension of a function in Fproj[g] to Gind[g]

as in Theorem 5.10.

For each a,b E g and 8 > 0 there exists ÀO > O, such that

Proof:

Let a,b E g and let E > O. Since g is a conic Köthe set, there exists

c E g such that c ~ a,b. We will prove the strenger statement: There

exists ÀO > O, such tbat

vx,yEC : jG(c·x + À0c•y) - G(c·x>I < 8.

2 Put y = 118c Gii F' For all À > 0 we have

$ yjexp(x+ Ày,x+Ày) 2 - exp(X+ Ày,x) 2 +

- exp(x,x + Ày)2

+ exp(x,x>2i $

$ yjexp(x+ Ày,x)2

j • jexp(x+ Ày,Ày)2

- 1 j +

+ yjexp(x,x>2i • jexp(X,Ày) 2 - 1 j $

1+À À+À2 .À $ ye (e - 1) + ye (ë - 1) •

So, it is clear that we can choose ÀO so small that

jG(a•x·+ À0b•y) ~ G(a·x>I < E for all x,y E C

Clearly ÀO depends on Il Se G Il F'

s. 12. Theorem + Let g c w (ID) be a type II conic Köthe set and let G be the extension

of a function in F .[g] to G. d[g] as in Theorem 5.10. pro3 1n

Then the function G is analytic on G. d[g]. 1n

0


Proof:

Let u E Gind[g] and let E > O, We prove that Gis continuous at u. We

assume that g ~ {am 1 m E lN} with an ;;.; an+t'

Let C denote the closed unit ball in .11,2 (ID). Since there exists b Eg -1

such that b • u E C, by Lemma 5.11 there exists À1

> 0 such that ·

VxEC : IG...1 a 1 • x) - G(u) 1 '< c./2 •

Inductively, we define a sequence (Àn)nElN in IR+:

149

Assume that we have found numbers À1, ••• ,Àm > 0 such that for ;;,; k;;,; m

(5. 1)

whe!te

v v : IG(u+ y+ À. ak• x) - Q(u+ y) 1 '< E/2k , xEC yEBk-l "k

k-1

~-1 = .l J=l

À. a .• C J J

k > 1 and {O} •

Put B = l~ 1 À.a.•C. Since gis a conic Köthe set, there exists C Eg, m J"' J J

such that u + Bm c: c • C. By Theorem 5. 11 there exists Àm+l > 0 sucb that

"xEC VyEB : iG(u+y+Àm+lam+1·x) - G(u+y)i < c./2m+1 • m

So we have found a sequence (Àn)nElN which satisfies (5.1) for all k.

Put V =co( U À a • C). Let v EV. Then there exist .\', E 1N, i3 > O, nElN n n .\', .\', m

x E C, 1 ;;,; m ;;,; R., such that \X. 1

B = 1 and v = \"' 1

i3 À a • x • m Lui= m L.m= m m m m

A simple application of the triangle inequality yields

IG<u+v) - GCu>I ~

R,

IG( u + ~ ) ( n-1 ;;,; l i3m Àa•x -Gu+l: B À a • x )1 n=1 m=1 m m m m=l m m m m

t ;;,; l c./2n :i e •

n=1

The set V is a convex, balanced subset of G. d[g], such that in

"vEV : IG(u+ v) - G(u) 1 < e •

For each n E lN the set V n i 2 [an;ID] is a neighbourhood of 0 in

;;,;


t 2 [an;ID], so the set V is a neighbourhood of ~ in Gind[g]. Hence the

function G is continuous at u.

So fora type II conic KÖthe set g, the space F .[g] can be identified proJ

with a subspace of A(Gind[g]).

The space F . [g] bas the same structure as the projective limits prOJ

introduced in Section 2. Theorem 4.1 states that in general the space

Seq[F . [a]] is a proper subspace of t+2

[up[a];lM(ID)]. Nevertheless, prOJ

Seq[F . [g)] and G . [up[g)] are the same. This statement is similar proJ proJ

to the corresponding statement for Find[g) in Theorem 5.6.

s. 13. Theorem + Let g c w (ID) be a conic Köthe set.

Then the operator Seq maps F .[g) homeomorphically onto G .[up[g]]. proJ proJ

Proof:

Let F E F . [g]. For all a E g we have 9 F E F and hence Seq [9a F] = proJ a

= up[a] ~ Seq[F] E t2

(lM(ID)). So Seq[F] EG . [up[g]].

a

prOJ -l Conversely, let f EG .[up[g]]. For each a Eg put Ga= Seq [up[a]•f].

proJ Then Ga is an element of F and certainly analytic on \l)(ID). The restric-

tions of the functions 9a-1 Ga and 9b-1 Gb to 11a • 11b• \P(ID) are equal,

so the following definition makes sense:

The function F on \l)(ID) is defined by F(X) = Ga(a-1 •x), where a Eg with

ID[x] c ID[a]' x E \l)(ID).

It is clear that F is ray-analytic on \P(ID) and hence analytic on \l)(ID)

and Seq[F] = f. Since 9 F = G E F, we get F E F .[g]. Since for all a a proJ

F E F .[g) and a Eg proJ .

118a FllF·= llup[a] •_Seq[F]llR, (M(ID)) , 2

the operator Seq maps F .[g] homeomorphically onto G .[up[g]]. a proJ proJ

In the final part of this section we combine the inductive limit and the

projective limit viewpoint.

The Theorems 5.6 and 5.13 state that the spaces G. d[up[g]] and in

G .[up[g]), both contained in w(:M(ID)), are representations of the proJ


spaces F. d[g] and F .[g], respectively, induced by the operator Seq. Ln proJ . These representation theorems have many topological consequences. Before

we summarize them, we introduce a pairing between the compound spaces

F. d [g] and F . [g ]. Ln proJ

5. 111. Definition + Let g c w (ID) denote a conic Köthe set and let (IDq)qElN be an exhaus-·

tion of ID.

The pairing <•,•>F between F. d[g] and F .[g] is defined by Ln proJ

<F ,G>f = lim '!T-q qElN

We note that qi(ID) and ~q are homeomorph. Cf. Corollary II.3.10. q

We remark that <F,G>F" <Seq[F],Seq[G]>, with <•,•>the pairing between

G. d{up[g]] and G .[up[g]] defined in (2.1). Ln proJ

We summarize the topological consequences of the representation theo

rems. We impose increasingly stronger conditions on the. set g.

(5.2) + Let g c w (ID) denote a conic Köthe set,

a. The space F. d[g] is barre led and bornological. Ln

b. The space Fproj [g] is complete.

(5.3) Let g denote a generating cone.

a. The inductive limit topology of F. d[g] equals the topology generated Ln by the seminorms ~, b. E (up [g ])# , defined by

'1, (F) " 1 b • Seq (F] 12 , F € F ind [g]

b. The space F. d[g] is nuclear iff Lll


c. We regard the elements of F. d[g] as analytic functions on G .[g]. in proJ Let F E f. d[gJ and let B denote a bounded set in G .[g]. Let

ln # prOJ a E g such that 0 _ 1 F E F and let b € g such that B c b • C, where

a C is the unit ball in Jl,

2 (ID). Then

2 sup IF(z) 1 :i 110 _ 1 FllF exp(i la•b 1..) • zEB a

The inductive limit topology of Find[g] is finer than the compact

open topology of Find[g]. Lemma I.2.3 implies that the seminorms

F 1+ sup IF(z) 1 , F € Find[g) , ze:B

with B a compact set, are continuous with r~spect to the inductive

limit topo logy.

d. The space F1.nd[g] is a representation of the strong dual of F .[g]. prOJ

e. The space F .[g] is semi-Montel iff prOJ

(5.4) Let g denote a #-symmetrie generating cone.

We regard the elements of F .[g) as ray-analytic functions on G1.nd[g]. prOJ

Let F € F .[g] and let B denote a bounded set in G1.nd[g]. Let a € g proJ

such that B c a • C. Then

So, the pré>jective iimit topology of Fproj[g] is finer than the compact

open topology of F .[g]. · proJ .

(5.5) Let g denote a generating cone such that up[g] is #-symmetrie.

A sufficient condition for this #-symmetry is the condition that g is

type II. Necessary is the condition that g is #-symmetrie.

a. The space Find[g] is complete and reflexive.

b. The space Fproj[g] is barreled, bornological and reflexive.


c. The space F .[g] is a representation of the strong dual of f1.nd[g].

proJ

d. If VaEg VbEg#: a•b € c0 (ID), then both spaces Find[g] and Fproj[g]

are Montel.

e. If vaEg vbEg# are nuclear.

a•b € Jl1(ID), then both spaces F. d[g] and F .[g] in proJ

Appendix

As an illustration of the theory developed in this chapter we present

representations of modified Heisenberg. groups in compound spaces.

The classical (2n+1)-dimensional Heisenberg group .l1i is the Lie group

IR.n x IR.n x IR.[mod 27r] with group operation

The classical unitary representation UH of ~H in L2(IR.n) is given by

(A.2) iy+i(X,W)z

[UH(v,w,y)F](x) == e F(v+ x) •

The modified Heisenberg groups are groups with a group operation and a

representation, which are very similar to the group operation • and the

representation UH of the classica! (2n+1)-dimensional Heisenberg group

~H.

Consider the following sketch.

Let V and W be two vector spaces which are in duality with respect to

the sesquilinear form<•,•>. The group H(V,W) is the set V x W x t witt

the group operation

Cf. (A.1)

For each (v,w,y) € H(V,W) and each complex valued function F on V we

define the complex valued function U(v,w,y)F on V by

(A.4) [U(v,w,y)F](x) == ey+<x,w> F(x+ v) •


Cf. (A.2). We have

So, if F0 is a space of functions on V, such that U(v,w,y) maps F0 into

F0, then U is a representation of H(V,W) in F0•

Consider the following operatörs:

(A.5) The translations Tv' v E V, with

[Tv F](x) • F(v + x) •

(A.6) The shifts Rw• w E W, with

CRw F]{X) = e<x,w> F(X) .

(A.7) The scalar multiplications Ey• y E C, with

[Ey F](x) = eY F(x) •

Then we have U(v,w,Y) = Eyo Rw 0 Tv.

Starting with a fixed conic Köthe set g c w+(:ID), we consider the

following two cases:

(A.8) CASE I:

(A. 9) CASE II :

F0 = F . [g]. proJ

We wil! prove that the translations TV and the shifts Rw are homeo

morphisms on F. d[g] and F .[g] for suitable V and w. The partial l.n prOJ

ordering of reproducing kemels is an essential tool in our proofs.

A.1. Lemma

Let u € Jl,2(ID). The complex valued function N(u) on qi(ID) x qi(ID) is

defined by N(u)(x,y) = exp[(x,u) 2 + (u,y) 2], x,y E qi(ID).


Then N( ) is a function of positive type and for all À > 0 there exists u À11

y > 0 such that N(u) ~ yK

Proof:

Let À > 0. For x,y E (j)(ID) we have

R k (II 1 4) . l" h N < KÀll "th (1 !2/À2) emar • • i.mp i.es t at (u) ~ y wi. y = exp u 2 •

A. 2. Corollary

Let u E i 2(ID) and let À> 1.

a. The function (x,y) 1+ exp(u+x,u+y) 2 belongs to PT((j)(ID)).

b. For each À > 1 there exists y > 0 such that

Proof:

a. Let ,Q, E 1N and let aj E 4:, Xj E (j)(ID), 1 ~ j ~ ,Q,. Then

,Q,

I akaj exp(U+Xk,U+XJ.)2 k,j=t

0 .

b. By LeDmla A. 1 there exis ts 8 > 0 such that N ( U) ~ 8 KIA-T 11 •

LeDmla II.t.20 implies that

lu1 2

with y = 8e 2

a

a


CASE I

We consider the group G .. d = H(G .[g],G. d[g]) and its represen-proJ,in pro) in tation (v,w,y) 1+ Ey~ Tv, further denoted by U, in the compound space

F. d[g]. Theorem 5.5 implies that we can extend the elements F of in F. d[g] to analytic functions on G .[g] in precisely one way. In the in proJ sequel we regàrd the functions F in F. d[g] both as analytic functions in on G • [g] and as analytic functions on (j)(lD).

prOJ

We start with a study of the translations T, v EG .[g], and the V prOJ

shifts R , WEG. d[g], on F. d[g]. For each F E F. d[g] the functions "'W in in in Tv F and ~ F are well defined on (j)(lD), Our first step is to prove that

the operators TV and ~ map elementary spaces continuously into elemen

tary spaces.

A. 3. Theorem

Let a,b E w+ (ID), v E .e.;ca,ID]' w E R,2 [b;ID].

a. For all À> 1, the translation Tv maps Find[a] continuously into

F. d[Àa]. ln

b. The shift ~ maps Find[a] continuously into Find[a+b],

Proof:

a. Let F E F. d[a]. Theorem 4.6 implies that we can regard F as a func-in +

tion on i;ca;ID] such that F(X) = F(11a • x) for all x E t 2 [a;m]. ·

Using Lemma II.1. 7 we prove that Tv F belongs to Find[Àa]. Let !/, E JN,

a. Et, x. E (j)(ID), 1 ~ j ~R.. Since 0 -1 F can be extended toa J J a

unique element of S(!l2(ID)), we get

R. 2 R, 1 I et. [Tv F](x.) 1 • 1 I a. F(V + x.) 12

= j=1 J J j•1 J J

Il,

lj!1

aj[ea_1 FHa•v+a•xj>l2

:;;

!/,

~ 110 _1 Fll~ l exp(a•v+ a•xk' a•v+ a•xj)Z • a k,j==1


Let À > t. Sincec a • v E i2

(ID), by Corollary A.2 there exists y > O,

depending on a • v, such that

R, 2 2 R, 2 1 l: et. [T F](x.) 1 ;; y • 110 _1 FN l: exp[À (a•xk.a•xJ.) 2 ] • j=t J v J a f k,j=l

Hence Tv F E find [Àa] and

b. Let F E Find[a]. For all x E q>(lp) we have ['\ F](x) = exp(<x,w>) •F(x).

The function x i+ exp(<x,w>), x E q>(ID), belongs to f. d[b]. in Theorem 5.2 yields that '\ F belongs to find[a+b] and

111\ Fllf. [a+b] ;?; exp(llb-1

• wl~) • llFllf. [a] • ind ind

a

A. 4. Corollary

Let g be a conic Köthe set, V EG .[g] and WEG. d[g]. prOJ . l.n The translation Tv and the shift Rw are homeomorphisms from find[g] onto

F. d[g]. in

Proof:

Since g is a cone, the translation Tv. maps f. d[g] into f. d[g] and for . l.n in . each a Eg the restriction TvlF· [a] is. continuous. Lemma I.2.3 yields ind _

1 that Tv is continuous on find[g]. Further, (Tv) " T_v• So the trans-

lation Tv is a homeomorphism from F. d[g] onto f. d[g]. in in The proof for the shift '\ runs similarly. a

A.s. Corollary

Let g be a conic Köthe set.

Then U represents the group G . . d as a group of continuous linear proJ,l.n operators on Find[g].

In the appendix to Chapter II we have introduced for each u E H the

annihilation operator a(u) and the creation operator y(u) in S(H). For

all suitable F E S(H) and x E H we have


[a(u)F](x) = lim (F(x +Au) - F(u))/;\ , ;\-!{)

[y(u)F](x) = (x,u)H F(x)

Moreover, we have seen in Appendix II.A that exp(a(u)) = T and u

exp(y(u)) = R • u

Tbis bas been our motivation to introduce for v € G .[g] and proJ w E Gind[g] the operators a(v) and y(W) on Find[g] by

[a(v)FJ(x) = lim (F(x + Av) - F(x))/À , À--+Ü

[y(W)F](x) = <X,W> F(x) •

It.can be proved that a(v) and y(w) are continuous operators from

Find[g] into Find[g]. The proof runs the same as in the cases of Tv and

~· We remark that a(V) is the inf initesimal generator of the group

{TÀV 1 À € t} and y(W) is the infinitesimal generator of the group

{RÀW 1 À € 4:}.

CASE II

We consider the group G. d . = H(G. d[g],G .[g]) and its represen-in ,proJ in proJ tation (v,w,y) 1+ E R Tv• denoted by U, in F . [g]. Theorem 5.10 states

. y·• ~~ that each F € F .[g] bas a unique ray-analytic extension G to G. d[g] proJ in such that for alla Eg the restriction G!t

2[a;ID] is analytic. In the

sequel we regard the functions in F .[g] both as ray-analytic funcproJ tions on G . [g] and as analytic functions on q>(ID). prOJ

We consider the translaÜons Tv• V E Gind[g], and the shifts ~· w € G . [g]. For each F € F • [g] the functions Tv F and R F are

proJ proJ ·• well defined on q>(ID).

A. 6. Theorem

Let g be a conic Köthe set, V € G. d[g] and W € G .[g]. ln proJ The translation Tv and the shift~ are homeomotphisms from Fproj[g]

onto F . [g]. prOJ


Proof:

Let F € F .[g]. By definition the.function T F belongs to F .[g] proJ . v . proJ

iff for all a E g the function 0 [T F] belongs :to F. a v _1

So, let a E g. There exists b E g such that b ~ a and b • v E i 2 (ID).

We use Lemma II.1.7 in order to prove that 0a[Tv F] E F. Let À> 1, R, € JN and let aj E 4:, xj E (f)(ID), 1 :il j :il R,. Put

R, 2 cr = 1 t aJ. ea [Tv F](xJ.) 1 •

j=1

Then

R, 2 jl, . 2 cr = 1 t aJ. [Tv F](a•xJ.) 1 = 1 t a. F(v + a•x.51 =

j=1 j=1 J J

R, . -1 -1 -1 -1 . 2 l Z: aJ.[0Àb F](À b •v +À b •a•xj>I :il j=1

-1 By Corollary A. 2 there exists y > 0, depending on À b • v, such that

b-1·a Because K :il K, we find

Hence 0a [Tv F] E F and

We conclude that the function Tv F belongs to fproj[gl and that the

translation TV maps F · .[g] continuously into F .[g]. Since _1

prOJ proJ . (Tv) = T-v' the translation Tv is a hom.eomorphism.

f60 IV Spaces of analytic functions

Let F E Fproj[g]. The function x >+ exp(<x,w>), x € ~(1D), belongs to

Fproj [g]. Theorem 5.9 implies that !\, F E Fproj [g] and

The remaining part of the proof runs ·similarly as in the previous case. c

A. 7. Corollary

Let g be a conic Köthe set.

Then U represents the group G. d . as a group of continuous linear i.n ,proJ operators on F . [g]. prOJ

As before, we define the operators a(v), v EG.- d[g], and y(w), in

w E Gproj[g] by

[a(V)F](x) lim (F(x + ÀV) - F(x))/l. , À>O

[y(w)F](x) = <x,w> F(x) ,

x E G • d [g] , F E F . [g] • l.n prOJ

They are continuous operators from F .[g] into F .[g] and they are proJ proJ the infinitesimal generators of the groups {î ÀV 1 À E C} and

{RÀW 1 À E C}, respectively.

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[Ku] Kuijlaars, A.B.J., Rijencollecties en #-symmetrie, Master's thesis, Eindhoven University of Technology, Eindhoven, 1987.

[Ma 1] Martens, F ,J .L., Function spaces of harmonie and analytic functions in infinieely many variables, EUT-Report 87-WSK-02, Eindhoven University of Technology, Eindhoven, 1987.

[Ma 2] , On the nuclearity of the spaces S~(A) and T~(A); Proc. K.N.A.W., A 89 (4) (1986), pp. 443-448.

[MOS] Magnus, W., F. Oberhettinger and R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 52, Springer-Verlag, Berlin etc., 1966.

[Mu] Milller, C., Spherical Harmonies, Leet. Notes in Math. 17, Springer-Verlag, Berlin etc., 1966.

[Na] Nachbin, L., Topology on Spaces of Holamorphic Mappings" Erg. der Matb. 47, Springer-Verlag, Berlin etc., 1969.

[Ru] Ruckle, W.H., Sequence Spaces, Research Notes in Math. 49, Pitman, Boston etc., 1981.

[Schl Schaefer, H.H., Topological Vector Spaces, G.T.M. 3, Springer Verlag, Berlin etc., 1986.

163

[Se] Segal, I.E., The Complex-Wave Representation of the Free Boson Field, Topics in Functional Analysis, Academie Press, New York, 1978, pp. 321-343.

[tE] Elst, A.F.M. ter, Distribution theories based on representations of locally compact Abelian topological groups, EUT-Report 88-WSK-02, Eindhoven Univers.ity of Technology, Eindhoven, 1988.

[Vi] Vilenkin, N.Ja, Special functions and the theory of group representations, Translations of mathematical monographs 22, A.M.S., 1986.

[Yo] Yosida, K., Functional Analysis, Grundlehren der mathematischen Wissenschaften 123, Springer-Verlag, Berlin etc., 1980.

absorbing analytic function analyticity space annihilation operator asymptotically equivalent

balanced Bargmann space

of finite order of infinite order

barrel barre led bornological

canonical injection Cauchy net collection of seminorms

directed -equivalent collections -gènerating -separating -

complete compound space conic positive sequence s.et convex creation operator cross

operation symmetry

cross

dilatation directed

collection of seminorms positive sequence se~

dual strong -weak -

elen:entary space equivalent exhaustion

function of exponential type of positive type

functional Hilbert space

INDEX

2 15,18

103 71 80

2

55 58

5 5 5

6 5

3 3 3 3 5

141 81 2

71 81 82 82

125

74

3 81 4 5 5

Gegenbauer polynomial generating

collection of seminorms set

harmonie function Hartogs' theorem Heisenberg group Hilbert tensor product Hilbertian dual system

inductive limit limit topology system

Köthe dual set space topology

locally convex space barreled -bornological -complete -Montel -nuclear -reflexive -semi-Montel -

locally convex topology

Mean Value Theorem Montel moulding set multi-index

set multiplier (p-)

134 n-fold 3 Hilbert tensor product

66

3 114

61,69 17

24,153 37 97

7 7 7

125 84

125 125

2 5 5 5 5 6 5 5 2

62 5

84 13 13 84

11 symmetrie tensor product symmetrizer

45 48 48

normal 132 space

30 topology 29

125 125

nuclear locally convex space operator

positive sequence set conic -directed -quasi-directed separating -solid -type r. II, III

projective limit limit topology system

quasi-directed quas:t-ordering

ray-analytic function reflexive

6 6

3 ff. 5 5 5 5 5

81

8 8 7

81 80

17,18 5

representation continuous -irreducible -unitary -

reproducing kemel

second quantization quantized operator

semi-Montel separating

collection of seminorms positive sequence set

sequence space skeleton solid spectra! trajectory space symmetrie Fock space

trajectory space type r. II, III

up operation

weight sequence

165

42 42 43 42 29

74 74 5

3 81

121 114

81 106 50

103 81

116

114

INDEX OF SYMBOLS

T(V,pr) 3 Hl H 29

B(V ,W), B (V) 4 ( • • •) H 29

V' 4 L 29 y

v~. V' i3 5 PT(E) 30

Vc W 7 V(E,L) 30

lim ind v 7 H(E,K) 31 aEA a

lim proj v 8 K1 ~ K2 32 a€A a

w(lC) 9 T(H1 , ••• ,Hn) 37

Xm 9 y(1) 0 ••• @ y(n) 37

nc [u] n

n[u], 9 @ K. 38 j=l J

tl ' f) 9 K1 • K2 39

tl u• ou 9 exp[K] 39

I • 1 9 Tn(H) 45

u-1 10 Q, Q@n 45

Ij>(][) 10 S(n) 48

I • 100

10 Tsym(H) 48 n

I • 1 11 F (H) 50 p sym

<- •• ) 2 11 E -x 50

lM(Il). lMn (Il) 13 W(F) 50

S! 14 S(H), Sn(H) 51

as 14 s 53

A(W) 15, 18 Bq 55

n B(u;r) q 16 K 55 q

n oB(u;r) 16 B 56 q q,n

167

K 56 <• '•> 98 q,n

Lq, Lq,n 57 SH,A 103

CR-2' m.12 58 TH,A 103

B"', B 59 S4>(A) 106 "',n

K"'" K 59 Îq.(A) 106 oo,n

H q 61 p 9 Q(U) 112

sq-1 62 up[a), up[g) 116

da q-1 62 R-2[a;ID), 1;[a; ID J 121

a q-1 62 Gind[g], Gproj[gl 122

H q,n 64 sx 127

d q,n 65 Seq 127

Mq' Mq,n 65 ev[a], ev(g] 128

CÀ n 66 Ka 134

a.(u) 71 F. d[aJ l.n 135

y(u) 71 K, F 135

w+ (Il) 79 F. d[g] 141 l.ll IP+ (l[) 80 Fproj[g] 144

~ 80 <•,•>F 151

80 H(V,W) 153

11:$. 80 N(U) 154

p# 81

Aa 85

Pa• 'la 86

Dom(-) 86

H(a] 87

I [p; (Hk)k€n], I[p;Hk] 88

H+[a] 93

P[p; (Hk) kE:D: l, P[p;Hk] 94

SUMMARY

This thesis is a treatment on spaces of analytic functions on sequence

spaces. Both the sequence spaces and the analytic function spaces belong

to a class of locally convex spaces, which are inductive limits of

Hilbert spaces, projective limits of semi-inner product spaces or both

at once.

By means of these analytic function spaces the concept of symmetr-ia

Fock 8pace for Hilbert spaces is generalized to sequence spaces in our

class. In this way, natural Fock space constructions can be carried out

for e.g. tèst spaces and distribution spaces.

Our treatise is an amalgamation of ideas from the theory of analytic

f.unctions on locally convex spaces, from the theory of the reproducing

kemel and from a recent study of inductive/projective limits of Hilbert

spaces.

As an application we construct representations of infinite dimensional

Heisenberg groups.

SAMENVATTING

Dit proefschrift gaat over ruimten van analytische functies op rijtjes

ruimten. Zowel de beschouwde rijtjesruimten als de ruimten van analy

tische functies behoren tot een klasse van lokaal convexe ruimten, die

inductieve limieten van Hilbertruimten, projectieve limieten van semi

inproduktruimten of beide tegelijk zijn.

Uitgaande van deze ruimten van analytische functies wordt het concept

symmetrische Fockruimte voor Rilbertruimten veralgemeend voor rijtjes

ruimten in onze klasse. Op basis van dit algemene resultaat kunnen dan

op een natuurlijke manier Fockruimte-constructies worden uitgevoerd

voor bijvoorbeeld testruimten en distributieruimten.

Onze beschouwingen zijn gebase.erd op ideeën uit de theorie van analy

tische functies op lokaal convexe ruimten, uit de theorie van reprodu

cerende kernen en uit een recente studie over inductieve/projectieve

limieten van Hilbertruimten.

Bij wij ze van toepassing worden representaties van oneindig dimensio

nale Heisenberggroepen geconstrueerd.

22 juni 1951

juni 1970

juli 1982

augustus 1983 tot

september .1987

vanaf september 1987

CURRICULUM VITAE

geboren te Lith

eindexamen gymnasium-S Bisschoppelijk College

te Roermond

doctoraal examen wiskundig ingenieur T.H.E.,

met lof

wetenschappelijk assistent T.H.E., vakgroep

analyse

medewerker bij de groep Basisonderwijs,

faculteit Wiskunde en Informatica, T.U.E.

STELLINGEN behorende bij net proefschrift

SPACES OF ANAL YTIC FUNCTIONS ON lNDUCTIVEIPROJECTIVE LIMITSOF HILBERT SPACES

door F.J.L. Manens

L De stelling van Kleinecke-Shirokov. 1 K 1, heeft de volgende asymptotische uitbreiding~

Zij A een compacte operator op een Hilbcrtruimte H_

Voor ledere e > 0 bcMaat er een ~ > 0 zodar1ig dat voor alle B " B(H) met 11 [A , fA , 8 :1 ,111 kleiner <ian S /11811 de spectrale straal van [A. 8 J kleiner dan t is.

2. Beschouw het Gelfandtripd

SL,(S,l.•~ '=--> L,(So) C...., T~,(S,).o::O

van respectievelijk de analytische futiCties, de kwadratisch integreerbare functies en de hyperfuncties op de eenheidssfccr S2 in !R 3 _ VgL [ G 1. Zij[, de oplossing van het Dirichletproblcem op de eenheidsbol met randvoorwaarde f" L,(s,,_, ), Zij I' de projectie in IR 3 met f'(..: 1 , x2 , x3)" (x 1 • D. DJ-

De operator /.,p met ll.r f 1 (i;)= J.(P ~) beeldt SL,(S,),d)l, continu in SL,(.>,).•~ af, maar i.~

niet uit te breiden tot een continue operator van Tus,) .•• ~ in 'f'L,(.ç,),t.!'..

3. Zij !>2(Sq_1) de Bilhertruimte van kwadratisch integree•·bare functies op de eenheidssfeer s •. 1 in JR• _Zij voorts SL(IR, q) de groep van reële q x 'f matrices met determinant 1. De afbeelding ot>, gegeven door het voorschrift

[~A)fl(~)= IA~ 1~" { 1:{12], l;e .'iq-I.[.;; L,(S •. ,).A" SL(IR, q),

is een unitaire representatie vatl SI.( IR, q) in L2(s._1).

Deze represetltatie is in-educibel op zowel de even als de oneven dervlruimtc in Lz(Sq-1)-

De operatoren tl>(A) zijn uit te breiden tot continue bijccties op de ruimte van hyperfuncties op s,_,'

4. Laat A een positieve :>:elfgeadjungeerde operator op 12 zijn met de eigenschap dat e-•A een Hilben-Schmidt operator is voor alle 1 > 0 en C de positieve zclfgcadjungccrdc operator op de Bargmannruimte B~ met ,__,c F"" F o •-'", FE B~, 1 > 0. De ruimte s~_.c bestaat uit precies die analytische functies F van T1,.t. in (f met de

eigenschap

3,.0 3">Q 'i_.,, : I F(z) I :;; ~ exp (~ I ,-•A z I i).

De duale ruimte T&..c bestaat uit precies die analytische functies F van s,.,, in I[

met de eigenschap

0',,0 ~,.,.o 0',,1, : I F(e-IA z) I :;; ~ exp <t I z I~).

Zie[M].

5. Beschouw de HUbertruimte Hz(B") van kwadratisch integreerbare harmonische functies op s,. de eenheidsbal in m•. De reproducerende kern K van H21Jl.l is

(I-I x Ij I y I~/ K(l,y)=Q (l+lx I~ ly lf:-(;:-y}z)f;+i-- (I+ I x I i ly 1~-(:<,yh)i-•.

De uitdrukking voor K, die vermeld wordt in [L], is niet correct.

6. Zij 1,, v > o, een semigroep van quasinilpotente operatoren op een Banachruimte x. Dan is voor iedere x eX, .t;<O, de verzameling {J,x I v > 0) onafhankelijk.

7. Zij S(IR") de Schwartzruirnte van snelafnemende functies. Zij A een dient gedellnieenle opel'ator in L 1(R") met de eigenschap datA"(S(IR")l c:S(IR"). Dan geldt

(a) A • is een continue afbeelding van de testruimte SC IR") in zichzelf_

(b) A i~ uit te breiden tot een continu~ afbeelding van de ruimte S'(IR") van getemperde distributies in zichzelf.

Vergelijk [E)_

8. Zij 1 een geheel analyti se he functie met positieve Taylorcoëfficiëntcn_ Dan is (-< , y) ,____, f((x , y )z!, x , y e 1 "' een reproducerende I<ern op 1, _ In de bijbehorende functionele Hilbcrtruimte liggen de harmonische functies van eindig veel variabelen overal dicht. Vergelijk Stelling ll.3.28 van dit proefschrift.

9. ledere vol nudeaire Kötheruimte is van de vorm C,""[l:l- Hierin is g een #symmetrische Köthevcrzamding.

Literatuur

IEl Eijndhoven, SJ.L. van, A theory of generalizcd functions based on one paran•eter gro~,~ps of unboundcd sclf-adjoint operators, TH-Report 81-WSK-03, Eindhoven University of Technology, Eindhoven, !981.

[G] Graaf,]_ de, Two spaces of generalizcd functions ba.sed on harmonie polynomials, in C. Brezinski, etc. (ed.), Polynömes orthogonaul\ et Applications, f>roc_ Bar-lc-Duc 1984, !eet. note~ in Math. ll7l. Springer-Verlag, 1984_

]KJ Kleinecke, D.C., On oper~tor commutators, Proc. A.M.S. 8 (1957), blz. 482-486.

[L] Ligocka, E., On the reproducing kemel for hat-monic functions and !he space of Bloch harmonie functions oo lhe unit ball in IR", Stud. Math., 87 (1987), blz. 23-32.

[M] Mancns, f'.J.l.-., 1-'unction spaces of harmonie and analytic functions in infinitely many variables, 8UT report 87-WSK-02, Eindhoven Univcrsity of T(;chnology, Eindhoven, 1987.

Eindhoven, 8 november 1988

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