Results Math 72 (2017), 2101–2120c© 2017 The Author(s). This article is an open accesspublication1422-6383/17/042101-20published online September 20, 2017DOI 10.1007/s00025-017-0750-0 Results in Mathematics
Paley-Wiener Isomorphism OverInfinite-Dimensional Unitary Groups
Oleh Lopushansky
Abstract. An analog of the Paley-Wiener isomorphism for the Hardy spacewith an invariant measure over infinite-dimensional unitary groups is de-scribed. This allows us to investigate on such space the shift and mul-tiplicative groups, as well as, their generators and intertwining opera-tors. We show applications to the Gauss-Weierstrass semigroups and tothe Weyl–Schrodinger irreducible representations of complexified infinite-dimensional Heisenberg groups.
Mathematics Subject Classification. 46T12, 46G20.
Keywords. Hardy spaces of infinitely many variables, Harmonic analysison infinite-dimensional groups, Symmetric Fock spaces.
1. Introduction
The work deals with the Hardy space H2χ of square-integrable C-valued func-
tions with respect to a probability measure χ over the infinite-dimensionalunitary group U(∞) :=
⋃{U(m) : m ∈ N}, extended by unit 1, which irre-
ducibly acts on a separable complex Hilbert space E with an orthonormalbasis {em}. Here, U(m) is the subgroup of unitary m × m-matrices endowedwith Haar’s measure χm.
In what follows, U(∞) is densely embedded via a universal mapping πinto the space of virtual unitary matrices U = lim←−U(m) defined as the projec-tive limit under Livsic’s mappings πm+1
m : U(m + 1) → U(m). The projectivelimit χ = lim←−χm, such that each image-measure πm+1
m (χm+1) is equal to χm,is concentrated on the range π(U(∞)) consisting of stabilized sequences (see[18,20]). The measure χ is invariant under right actions [20, n.4].
2102 O. Lopushansky Results Math
We refer to [5,26] for applications of χ to stochastic processes. Neededproperties of Hardy spaces H2
χ can be found in [15]. Various cases of Hardyspaces in infinite-dimensional settings were considered in [9,17].
Now, we briefly describe results. Using a unitarily weighted symmetricFock space (Γw, 〈· | ·〉w) with a canonical orthogonal basis of symmetric tensorproducts {e�λ
ı } of basis elements {em} ⊂ E indexed by Young diagrams λ andnormalized by measure χ, we find an orthogonal basis of polynomial {φλ
ı } inH2
χ such that the conjugate-linear mapping
Φ : Γw → H2χ
is a surjective isometry with one-to-one correspondence e�λı � φλ
ı . This allowsus to establish in Theorem 2 an integral formula for a Fock-symmetric F-transform
F : H2χ f → f ∈ H2
w
where the Hilbert space H2w, uniquely determined by Γw, consists of Hilbert–
Schmidt analytic entire functions on E. Thus, the F-transform acts as ananalog of the Paley-Wiener isomorphism over infinite-dimensional groups.
Furthermore, we investigate two different representations of the additivegroup (E,+) over the Hardy space H2
χ by shift and multiplicative groups.Theorem 3 states that the F-transform is an intertwining operator betweenthe multiplication group M†
a on H2χ and the shift group Ta on H2
w. On theother hand, Theorem 4 shows that F is the same between the shift group T †
a
on H2χ and the multiplication group Ma∗ on H2
w. Integral formulas describinginterrelations between their generators are established. In Theorem 5 suitablecommutation relations are stated.
Applications to the Gauss-Weierstrass-type semigroups on H2χ are shown
in Theorem 6. Another application to linear representations of complexifiedinfinite-dimensional Heisenberg groups on H2
χ in a Weyl–Schrodinger form isgiven in Theorem 7.
Infinite-dimensional Heisenberg groups was considered in [16] by usingreproducing kernel Hilbert spaces. The Schrodinger representation of infinite-dimensional Heisenberg groups on L2
γ with respect to a Gaussian measure γover a real Hilbert space is described in [3] (see also earlier publications [1,2]).
In conclusion, we note that a motivation for this study was the follow-ing simple relations in the Hardy space H2
χ over 1-dimensional group U(1) ={u = exp(iϑ) : ϑ ∈ [−π, π]}. In this case, {un : n ∈ Z+} is an orthonormal ba-sis and Γw � 2, since Φ∗f = (fn) ∈ 2 for any f ∈ H2
χ with Fourier coeffi-cients fn =
∫f(u)undu = 〈un | Φ∗f〉w. While, f(x) =
∫f(u) exp(xu) du =∑
fnxn/n! = 〈ε(x) | Φ∗f〉w, where ε(x) = (xn/n!) ∈ 2 for all x ∈ C.Moreover, the equalities Taf(x) =
∫f(u) exp[(x + a)u] du = F(M†
af)(x)with M†
af(u) = exp(au)f(u) hold for all x, a ∈ C. On the other hand, (Maf)(x)= exp(xa)
∑fnxn/n! =
∑(T †
afn)xn/n! with T †afn =
∑nk=0
(nk
)akfn−k. Hence,
Vol. 72 (2017) Paley-Wiener Isomorphism 2103
Maf = F(T †af), where (T †
af)(u) =∑
(T †afn)un. In result, TaMb = exp(ab)
MbTa for all a, b ∈ C and the Weyl–Schrodinger representation of Heisenberg’sgroup from Theorem 7 retains a classic form.
The case H2χ over m-dimensional group U(m) is similar with a proviso
that the weighted Fock space Γw is normalized by ‖e�λı ‖w =
(n+m−1
n
)−1/2
where n = |λ| is a homogeneity degree of the basis polynomial φλı in H2
χ.Note that the normalization ‖e�λ
ı ‖w = n!−1/2 with n = |λ| leads to the caseof Segal-Bargmann’s space H2
γ with standard centered probability Gaussianmeasure γ on C
m.
2. Hilbert–Schmidt Analyticity
Let E stand for a separable complex Hilbert space with scalar product 〈· | ·〉norm ‖ · ‖ and a fixed orthonormal basis {ek : k ∈ N}. Denote by E⊗n
alg =
E⊗ n times· · · ⊗E (n ∈ N) its algebraic tensor power consisted of the linearspan of elements ψn = x1 ⊗ · · · ⊗ xn with xi ∈ E (i = 1, . . . , n). Set x⊗n :=
x⊗ n times· · · ⊗x. The symmetric algebraic tensor power E�nalg = E � · · · � E is
defined to be the range of the projector sn : E⊗nalg ψn → x1 � · · · � xn with
x1 � · · · � xn := (n!)−1∑
σxσ(1) ⊗ · · · ⊗ xσ(n) where σ : {1, . . . , n} → {σ(1),. . . , σ(n)} runs through all permutations. The symmetric algebraic Fock spaceis defined as the algebraic direct sum Γalg =
∑n∈Z+
E�nalg with E�0
alg = C.
Let E⊗nh := E ⊗h · · · ⊗h E be the completion of E⊗n
alg by Hilbertian norm
‖ψn‖h = 〈ψn | ψn〉1/2h with 〈ψn | ψ′
n〉h = 〈x1 | x′1〉 · · · 〈xn | x′
n〉. Denote by E�nh
the range of continuous extension of sn on E⊗nh . As usual, the symmetric Fock
space is defined to be the Hilbertian direct sum Γh =⊕
n∈Z+E�n
h .Denote by λ = (λ1, . . . , λm) ∈ Z
m+ with λ1 ≥ λ2 ≥ · · · ≥ λm a partition
of n ∈ N, that is, n = |λ| where |λ| := λ1 + · · · + λm. Any λ may be identifiedwith Young’s diagram of length l(λ) = m. Let Y denote all diagrams andYn = {λ ∈ Y : |λ| = n}. Assume that Y0 = {∅ ∈ Y : |∅| = 0} and l(∅) = 1. LetN
m∗ := {ı = (ı1, . . . , ım) ∈ N
m : ıl =/ ık, ∀ l =/ k}. For each λ ∈ Y we assign theconstant
C|λ|,l(λ) :=(l(λ) − 1)!|λ|!
(l(λ) − 1 + |λ|)! ≤ 1. (1)
The spaces E�nalg and Γalg may be generated by the basis of symmetric
tensors
e�Yn =⋃{
e�λı := e⊗λ1
ı1 � · · · � e⊗λl(λ)ıl(λ) : (λ, ı) ∈ Yn × N
l(λ)∗
},
e�Y =⋃{
e�Yn : n ∈ Z+
}with e�∅
ı = 1,
2104 O. Lopushansky Results Math
respectively. As is known [4, Sect. 2.2.2], norm of basis element in Γh is equalto
‖e�λı ‖2h =
λ!|λ|! , λ! := λ1! · · · λm!. (2)
Let us define a new Hilbertian norm on Γalg by the equality‖ · ‖w = 〈· | ·〉1/2
w where scalar product 〈· | ·〉w is determined via the orthogonalrelations
〈e�λı | e�λ′
ı′ 〉w ={
C|λ|,l(λ)‖e�λı ‖2h : λ = λ and ı = ı′,
0 : λ =/ λ′ or ı =/ ı′.
Denote by E�nw and Γw the appropriate completions of E�n
alg and Γalg, respec-
tively. For any ı ∈ Nl(λ)∗ there corresponds in E�n
w the d-dimension subspacewith d = C−1
|λ|,l(λ), spanned by elements{e�λı : λ ∈ Yn
}. The Hilbertian or-
thogonal sum
Γw =⊕
n∈Z+
E�nw
endowed with 〈· | ·〉w we will call unitarily weighted symmetric Fock space.Let x =
∑ekxk be the Fourier series of x ∈ E with coefficients xk =
〈x | ek〉. We assign to any (λ, ı) ∈ Yn × Nl(λ)∗ the n-homogenous Hilbert–
Schmidt polynomial defined via the Fourier coefficients
xλı := 〈x⊗n | e�λ
ı 〉w = xλ1ı1 . . . x
λl(λ)ıl(λ) , x ∈ E.
Using the tensor multinomial theorem, we define in Γw the Fourier decompo-sition of exponential vectors (or coherent state vectors)
ε(x) :=⊕
n∈Z+
x⊗n
n!=
⊕
n∈Z+
1n!
(∑
k∈N
ekxk
)⊗n
=⊕
n∈Z+
1n!
∑
(λ,ı)∈Yn×Nl(λ)∗
n!λ!
e�λı xλ
ı
(3)
with respect to the basis e�Y. It is convergent in Γw in view of (1) and
‖ε(x)‖2w =∑
n∈Z+
1n!2
∑
(λ,ı)∈Yn×Nl(λ)∗
(n!λ!
)2‖e�λ
ı ‖2w|xλı |2
=∑
n∈Z+
1n!2
∑
(λ,ı)
n!λ!
C|λ|,l(λ)|xλı |2 ≤
∑
n∈Z+
1n!
∑
(λ,ı)
n!λ!
|xλı |2
=∑
n∈Z+
1n!
(∑
k∈N
|xk|2)n
= e‖x‖2.
(4)
Particulary, (4) implies that the function E x → ε(x) ∈ Γw is entire analytic.
Vol. 72 (2017) Paley-Wiener Isomorphism 2105
Definition 1. The space of C-valued Hilbert–Schmidt entire analytic functionsin variable x ∈ E, associating with the unitarily weighted symmetric Fockspace Γw, is defined to be
H2w := {ψ∗(x) := 〈ε(x) | ψ〉w : ψ ∈ Γw} with the norm ‖ψ∗‖ = ‖ψ‖w .
Every function ψ∗ is entire analytic as the composition of ε(·) with〈· | ψ〉w. The subspace in H2
w of n-homogenous Hilbert–Schmidt polynomialsis defined to be
H2,nw =
{ψ∗
n(x) = 〈x⊗n | ψn〉w : ψn ∈ E�nw
}.
Evidently, H2w = C ⊕ H2,1
w ⊕ H2,2w ⊕ . . ..
It is important that H2w is uniquely determined by Γw since {ε(x) : x ∈ E}
is total in Γw. Similarly, for the subspace H2,nw which is uniquely determined
by E�nw , since {x⊗n : x ∈ E} is total in E�n
w . The last totality follows from thepolarization formula for symmetric tensor products
e�λı =
12nn!
∑
θ1,...,θn=±1
θ1 . . . θn a⊗n with a =l(λ)∑
i=1
θie⊗λiıi
(5)
which is valid for all e�λı ∈ e�Yn (see e.g. [11, Sect. 1.5]) Thus, the conjugate-
linear isometries ψ → ψ∗ from Γw onto H2w and from E�n
w onto H2,nw hold.
In conclusion, we can notice that every analytic function ψ∗ ∈ H2w deter-
mined by ψ =⊕
ψn ∈ Γw, (ψn ∈ E�nw ) has the Taylor expansion at zero
ψ∗(x) =∑
n∈Z+
1n!
∑
(λ,ı)∈Yn×Nl(λ)∗
〈e�λı | ψn〉w‖e�λ
ı ‖2wxλ
ı , x ∈ E
that follows from (3). The function ψ∗ is entire Hilbert–Schmidt analytic [15,n.5].
Note that analytic functions of Hilbert–Schmidt types were also consid-ered in [10,14,21]. More general classes of analytic functions associated withcoherent sequences of polynomial ideals were described in [8].
3. Hardy Space Over U(∞)
In what follows, we endow each group U(m) with the probability Haar mea-sure χm and assume that U(m) is identified with its range with respect to
the embedding U(m) um →[um 00 1
]
∈ U(∞). The Livsic transform from
U(m + 1) onto U(m) is described in [18, Proposition 0.1] and [20, Lemma 3.1]as the surjective Borel mapping
πm+1m : um+1 :=
[zm ab t
]
−→ um :={
zm − [a(1 + t)−1b] : t �= −1zm : t = −1.
2106 O. Lopushansky Results Math
The projective limit U := lim←−U(m) under πm+1m has surjective Borel projec-
tions πm : U u → um ∈ U(m) such that πm = πm+1m ◦ πm+1.
Consider a universal dense embedding π : U(∞) � U which to everyum ∈ U(m) assigns the stabilized sequence u = (uk) such that (see [20, n.4])
π : U(m) um → (uk) ∈ U, uk ={
πmk (um) : k < m
um : k ≥ m,(6)
where πmk := πk+1
k ◦ . . . ◦ πmm−1 for k < m and πm
k is identity mapping fork ≥ m. On its range π(U(∞)), endowed with the Borel structure from U, weconsider the inverse mapping
π−1 : Uπ → U(∞) where Uπ := π(U(∞)).
The right action Uπ u → u.g ∈ Uπ with g = (v, w) ∈ U(∞) × U(∞) isdefined by πm(u.g) = w−1πm(u)v where m is so large that g = (v, w) ∈U(m) × U(m).
Following [18, n.3.1], [20, Lemma 4.8] via the Kolmogorov consistencytheorem (see e.g. [19, Theorem 1], [24, Corollary 4.2]) we uniquely define onU = lim←−U(m) the probability measure χ := lim←−χm such that each image-measure πm+1
m (χm+1) is equal to χm. For any Borel subset A ⊂ Uπ we haveπm+1(A) ⊆ (πm+1
m )−1 [πm(A)], because πm = πm+1m ◦ πm+1. It follows that
(χm ◦ πm)(A) = πm+1m (χm+1)[πm(A)] = χm+1[(πm+1
m )−1[πm(A)]] ≥ (χm+1 ◦πm+1)(A). Hence, χ satisfies the condition
χ(A) = inf(χm ◦ πm)(A) = lim χm(A) (7)
and therefore the projective limit lim←−χm exists on Uπ via the well knownProhorov theorem [6, Theorem IX.52]. Moreover, it is a Radon probabilitymeasure concentrated on Uπ [24, Theorem 4.1]. By the known portmanteautheorem [13, Theorem 13.16] and Fubini’s theorem the invariance of Haarmeasures χm together with (7) yield the following invariance properties underthe right action
∫
f(u.g) dχ(u) =∫
f(u) dχ(u), g ∈ U(∞) × U(∞), f ∈ L∞γ ,
∫
f dχ =∫
dχ(u)∫
U(m)×U(m)
f(u.g) d(χm ⊗ χm)(g),
where L∞χ stands for the space of all χ-essentially bounded complex-valued
functions defined on Uπ and endowed with norm ‖f‖∞ = ess supu∈Uπ|f(u)|.
Let L2χ be the space of square-integrable C-valued functions f on Uπ with
norm
‖f‖χ = 〈f | f〉1/2χ where 〈f | f〉χ :=
∫
f1f2 dχ.
The embedding L∞χ � L2
χ holds, moreover, ‖f‖χ ≤ ‖f‖∞ for all f ∈ L∞χ .
Vol. 72 (2017) Paley-Wiener Isomorphism 2107
To given the E-valued mapping Uπ u → π−1(u)e1, we can well-definethe Borel χ-essentially bounded functions in the variable u ∈ Uπ,
φk := φek, φek
(u) =⟨π−1(u)e1 | ek
⟩, k ∈ N,
which do not depend on the choice of e1 in⋃
S(m) where S(m) isthe m-dimensional unit sphere in E [15, n.3]. The uniqueness ofφx(u) = 〈π−1(u)e1 | x〉 with x ∈ E results from the total embeddingπ : U(∞) � U. From (6) it follows that π−1 ◦ π−1
m coincides with the em-bedding U(m) � U(∞). Hence, by (7) and the portmanteau theorem thereexist the limit
∫
φx dχ = limm→∞
∫
U(m)
φx d(χm ◦ πm) = limm→∞
∫
U(m)
(φx ◦ π−1m ) dχm,
i.e., φx ∈ L∞χ for any φx(u) = 〈π−1(u)e1 | x〉 with x ∈ E.
By formula (5) to every e�λı ∈ e�Yn there uniquely corresponds the Borel
function from L∞χ
φλı (u) :=
⟨[π−1(u)e1]⊗n | e�λ
ı
⟩w
= φλ1ı1 (u) . . . φ
λl(λ)ıl(λ) (u)
in the variable u ∈ Uπ. It follows that the orthogonal basis e�Y of elementse�λı = e⊗λ1
ı1 � · · · � e⊗λmım
, indexed by λ = (λ1, . . . , λm) ∈ Y and ı = (ı1, . . . , ım)∈ N
m∗ with m = l(λ), uniquely determines the systems of Borel χ-essentially
bounded functions in the variable u ∈ Uπ,
φYn =⋃{
φλı := φλ1
ı1 · · · φλmım
: (λ, ı) ∈ Yn × Nm∗ , m = l(λ)
},
φY =⋃{
φYn : n ∈ Z+
}with φ∅
ı ≡ 1.
Definition 2. The Hardy space H2χ is defined as the closed complex linear span
of φY endowed with L2χ-norm.
The following assertion is proved in [15, Theorem 3.2].
Theorem 1. The system of Borel functions φY forms an orthogonal basis inH2
χ such that
‖φλı ‖χ = C
1/2|λ|,l(λ)‖e�λ
ı ‖h, λ ∈ Y, ı ∈ Nl(λ)∗ .
Define the subspace H2,nχ ⊂ H2
χ for any n ∈ N to be the closed linearspan of the subsystem φYn . Theorem 1 implies that H2,n
χ ⊥ H2,mχ in L2
χ forany n =/ m. This provides the orthogonal decomposition
H2χ = C ⊕ H2,1
χ ⊕ H2,2χ ⊕ · · · .
2108 O. Lopushansky Results Math
4. Fock-Symmetric F -Transform
The one-to-one correspondence e�λı � φλ
ı allows us to define via the changeof orthonormal bases
Φ : Γw e�λı ‖e�λ
ı ‖−1w → φλ
ı ‖φλı ‖−1
χ ∈ H2χ, λ ∈ Y, ı ∈ N
l(λ)∗
the isometric conjugate-linear mapping Φ : Γw → H2χ. The adjoint mapping
Φ∗ : H2χ → Γw is defined by
⟨Φe�λ
ı | f⟩
χ=⟨e�λı | Φ∗f
⟩w
with f ∈ H2χ. The
suitable Fourier decomposition has the form
Φψ =∑
(λ,ı)∈Y×Nl(λ)∗
ψ(λ,ı)φλı ‖φλ
ı ‖−1χ , ψ(λ,ı) := 〈e�λ
ı | ψ〉w ‖e�λı ‖−1
w
for any ψ ∈ Γw. In particular, the equality Φx =∑
xkφk is valid for all x ∈ E.This gives the equalities
‖Φx‖2χ =∑
|xk|2 = ‖x‖2, x ∈ E.
Using this, we will examine the composition of Φ with the Γw-valued functionε : E x → ε(x). Its correctness justifies the following assertion that substan-tially uses the L∞
χ -valued function
φx : Uπ u → (Φx)(u) =∑
xkφk(u)
which is linear in the variable x ∈ E.Similarly to the known case of Wiener spaces, the function Φx can be
seen as a group analog of the Paley-Wiener map (see e.g. [12, n.4.4] or [23]).
Lemma 1. The composition Φε(x), which is understood as the function
[Φε(x)](u) : Uπ u → exp (φx(u)) ,
takes values in L∞χ for all x ∈ E.
Proof. Applying Φ to the Fourier decomposition (3), we obtain
Φε(x) =∑
n∈Z+
1n!
∑
(λ,ı)∈Yn×Nl(λ)∗
n!λ!
xλı φλ
ı =∑
n∈Z+
1n!
(∑
k∈N
xkφk
)n
= exp (φx) .
It directly follows that ‖Φε(x)‖∞ ≤ exp ‖φx‖∞. �
Theorem 2. For every f =∑
fn ∈ H2χ, (fn ∈ H2,n
χ ) the entire analytic func-tion f(x) := 〈ε(x) | Φ∗f〉w in the variable x ∈ E and its Taylor coefficients atorigin have the integral representations
f(x) =∫
exp(φx)f dχ and dn0 f(x) =
∫
φnxfn dχ, (8)
respectively. The mapping F : H2χ f → f ∈ H2
w (regarded as a Fock-symmet-ric F-transform) provides the isometries
H2χ � H2
w and H2,nχ � H2,n
w .
Vol. 72 (2017) Paley-Wiener Isomorphism 2109
Proof. First recall that the Γw-valued function ε(·) is entire analytic on E,therefore f is the same, as the composition of ε(·) with 〈· | Φ∗f〉w. Farther on,consider the Fourier decomposition with respect to the basis φY,
f =∑
n∈Z+
fn =∑
n∈Z+
(λ,ı)∈Yn×Nl(λ)∗
fλ,ı,nφλı
‖φλı ‖χ
, fλ,ı,n =1
‖φλı ‖χ
∫
f φλı dχ.
Applying Φ∗ to f in this decomposition and substituting fλ,ı,n into f , weobtain
f(x) =∑
n∈Z+
1n!
∑
(λ,ı)∈Yn×Nl(λ)∗
n!λ!
fλ,ı,n〈e�λı | e�λ
ı 〉wxλı
‖e�λı ‖w
=∫ ∑
n∈Z+
1n!
( ∑
(λ,ı)∈Yn×Nl(λ)∗
n!λ!
xλı φλ
ı
)f dχ =
∫
exp(φx
)f dχ
where the last equality is valid by Lemma 1. It particularly follows that fory = αx,
f (y) =∫
exp(φαx
)f dχ =
∑αn
∫φn
x
n!fndχ, α ∈ C.
Differentiating f at y = 0 and using the n-homogeneity of derivatives, weobtain
dn0 f(x) =
dn
dαn
∑αn
∫φn
x
n!fn dχ
∣∣∣α=0
=∫
φnxfn dχ.
Finally, we notice that the isometry H2χ � H2
w holds, since the isometry Φ∗ issurjective. In the case of polynomials we similarly get H2,n
χ � H2,nw . �
Note that a different integral formula for analytic functions employingWiener measures on infinite-dimensional Banach spaces was presented in [22].
5. Exponential Creation and Annihilation Groups
Let us define the linear mapping jn : E�nw → E�n
h to be the continuous ex-tension of identity mapping acting on the dense subspace E�n
alg ⊂ E�nw ∩ E�n
h .Such continuous extension jn is a contractive injection with dense range. Infact, it suffices to expand elements from E�n
w and E�nh into the Fourier series
with respect to orthogonal basis e�Yn and apply the inequality
‖e�λı ‖2w = C|λ|,l(λ)‖e�λ
ı ‖2h ≤ ‖e�λı ‖2h, λ ∈ Yn (9)
which follows from Theorem 1, taking into account the inequality (1). Us-ing subsequently that E�n
h is reflexive, we obtain that its adjoint operator
2110 O. Lopushansky Results Math
j∗n : E�nh → E�n
w is a contractive injection with dense range. Thus, the map-
ping jn is also injective. Moreover, E�nh
j∗n→ E�nw
jn� E�nh forms a Gelfand triple.
Particularly, the operator sn possesses continuous extension on E�nw .
Using this, we consider the linear operator
sn/m := sn ◦ (jm ⊗ jn−m) with m ≤ n
defined to be φm � ψn−m = sn/m(φm ⊗ ψn−m) ∈ E�nw for all φm ∈ E�m
w ,ψn−m ∈ E
�(n−m)w .
Lemma 2. The mapping sn/m from E�mw ⊗h E
�(n−m)w to E�n
w is a contractiveinjection with dense range.
Proof. Expand elements of E�mw ⊗h E
�(n−m)w with respect to e�λ
ı ⊗ e�μj for
all λ, μ ∈ Y, ı ∈ Nl(λ), j ∈ N
l(μ) such that |λ| = m, |μ| = n − m. Using (9), wehave
‖e�λı ⊗ e�μ
j ‖E�m
w ⊗hE�(n−m)w
= ‖e�λı ‖w‖e�μ
j ‖w≤ ‖e�λ
ı ‖h‖e�μj ‖h = ‖e�λ
ı ⊗ e�μj ‖h.
As above, it implies that the mapping jm ⊗ jn−m : E�mw ⊗h E
�(n−m)w → E⊗n
h ,defined to be the continuous extension of identity mapping on E�m
alg ⊗E�(n−m)alg ,
is a contractive injection. Using subsequently that E�mh ⊗h E
�(n−m)h is reflex-
ive, we get the Gelfand triple
E�mw ⊗h E�(n−m)
w
sn/m→ E�nh
j∗n→ E�nw
where injections are contractive and have dense ranges. �
Lemma 3. The Γw-valued function, defined on {ε(x) : x ∈ E} by
Taε(x) = ε(x + a),
has a unique linear extension Ta : Γw ψ → Taψ ∈ Γw such that
‖Taψ‖2w ≤ exp(‖a‖2)‖ψ‖2w and Ta+b = TaTb = TbTa for all a, b ∈ E.
Proof. Let us define the creation operators δma,n : E
�(n−m)w → E�n
w (m ≤ n) as
δma,nx⊗(n−m) := sn/m
[a⊗m ⊗ x⊗(n−m)
]=
(n − m)!n!
dm(x + ta)⊗n
dtm
∣∣∣t=0
(10)
for all a, x ∈ E. Note that the second equality in (10) follows from the bino-mial formula for symmetric tensor elements (x + ta)⊗n =
∑nm=0
(nm
)(ta)⊗m �
x⊗(n−m). Put δ0a,n = 1. If a = 0 then δm0,n = 0. Summing over n ≥ m with
coefficients 1/(n − m)!, we get
δma ε(x) =
dmε(x + ta)dtm
∣∣∣t=0
=⊕
n≥m
sn/m[a⊗m ⊗ x⊗(n−m)](n − m)!
, t ∈ C. (11)
Vol. 72 (2017) Paley-Wiener Isomorphism 2111
This series is convergent, since by Lemma 2 and (4) the inequality
‖δma ε(x)‖w ≤ ‖a‖m
∥∥∥⊕
n≥m
x⊗(n−m)
(n − m)!
∥∥∥w
= ‖a‖m ‖ε(x)‖w (12)
holds. From (11) and the tensor binomial formula mentioned above it followsthat
n⊕
m=0
1m!
δma,n
x⊗(n−m)
(n − m)!=
n⊕
m=0
a⊗m � x⊗(n−m)
m!(n − m)!=
(x + a)⊗n
n!.
Summing over n ∈ Z+ with coefficients 1/n! and using (11), we obtain
Taε(x) =⊕
n∈Z+
n∑
m=0
1m!
δma,n
x⊗(n−m)
(n − m)!
=∑
m∈Z+
1m!
⊕
n≥m
δma,n
x⊗(n−m)
(n − m)!= exp(δa)ε(x).
The inequalities (4) and (12) yield ‖Taε(x)‖2w ≤ exp(‖a‖2
)‖ε(x)‖2w. Taking
into account the totality of {ε(x) : x ∈ E}, this inequality implies the requiredinequality on Γw. It also follows that Ta+b = TaTb = TbTa, since δa+b = δa + δbfor all a, b ∈ E by linearity of creation operators. This ends the proof. �
We define the adjoint operators δ∗ma,n : E�n
w ψn → δ∗ma,nψn ∈ E
�(n−m)w as
⟨δma,nx⊗(n−m) | ψn
⟩w
=⟨x⊗(n−m) | δ∗m
a,nψn
⟩w, a, x ∈ E
for n ≥ m. It immediately follows that for every ψn−m ∈ E�(n−m)w and x ∈ E,
⟨δ∗ma,nx⊗n | ψn−m
⟩w
=⟨x⊗n | δm
a,nψn−m
⟩w
=⟨x⊗n | a⊗m � ψn−m
⟩w
= 〈x | a〉m⟨x⊗(n−m) | ψn−m
⟩w.
(13)
Using δ∗ma,n , we can uniquely define a Γw-valued function T ∗
a by the equalities
T ∗a ε(x) = exp(δ∗
a )ε(x) =∑
m∈Z+
δ∗ma ε(x)
m!, δ∗m
a ε(x) :=⊕
n≥m
δ∗ma,nx⊗n
n!(14)
for all a, x ∈ E. Taking into account Lemma 3, we obtain the following claim.
Lemma 4. The Γw-valued function T ∗a , defined by (14), possesses a unique
linear extension T ∗a : Γw ψ → T ∗
a ψ ∈ Γw such that
‖T ∗a ψ‖2w ≤ exp(‖a‖2) ‖ψ‖2w and T ∗
a+b = T ∗a T ∗
b = T ∗b T ∗
a for all a, b ∈ E.
Definition 3. We will call the Γw-valued functions Ta and T ∗a in variable a ∈ E
the exponential creation and annihilation groups, respectively.
2112 O. Lopushansky Results Math
6. Intertwining Properties of F -Transform
Let us define on the space H2χ the multiplicative group M†
a : E a → M†a to
be
M†a f(u) = exp[φa(u)]f(u), f ∈ H2
χ, u ∈ Uπ.
It can be considered as a linear representation of the additive group (E,+).By Lemma 1 the function u → exp[φa(u)] with a fixed a belongs to L∞
χ . Hence,M†
a is continuous on H2χ. The generator of the 1-parameter group C t → M†
ta
coincides with the operator of multiplication by the L∞χ -valued function
φa : Uπ u → φa(u) where dM†ta/dt|t=0 = φa.
The continuity of E a → exp(φa) implies that this 1-parameter groupM†
ta is strongly continuous on H2χ. As a consequnce, its generator (φaf)(u) =
φa(u)f(u) with domain D(φa) ={f ∈ H2
χ : φaf ∈ H2χ
}is closed and densely-
defined. As well, its power φma defined on D(φm
a ) ={f ∈ H2
χ : φma f ∈ H2
χ
}for
any m ∈ N is the same (see, e.g. [7] for details).The additive group (E,+) may be also linearly represented on H2
w as theshift group
Taf(x) = f(x + a), f ∈ H2χ, x, a ∈ E.
The directional derivative on the space H2w along a nonzero a ∈ E coincides
with the generator of the 1-parameter shift subgroup C t → Tta, that is,
daf = limt→0
t−1(Ttaf − f) with domain D(da) :={f ∈ H2
w : daf ∈ H2w
}.
Note that the 1-parameter shift group Tta, which is intertwined with M†ta by
the F-transform
Ttaf(x) =∫
exp[φx+ta
]f dχ =
∫
exp(φx)M†taf dχ, (15)
is strongly continuous on H2w. Since D(dm
a ) contains all polynomials from H2w,
each operator dma with domain D(dm
a ) ={f ∈ H2
w : dma f ∈ H2
w
}is closed and
densely-defined. From (15) it directly follows
dma f(x) =
∫
exp(φx)dmM†
ta
dtm
∣∣∣t=0
f dχ =∫
exp(φx)φma f dχ (16)
for all f ∈ D(φma ) and x ∈ E. On the other hand, by Theorem 2 we have
Taf(x) = 〈Taε(x) | Φ∗f〉 = 〈ε(x) | T ∗a Φ∗f〉 =
∫
exp(φx)ΦT ∗a Φ∗f dχ. (17)
Theorem 2 together with (15) and (17) imply that M†a is connected with the
exponential annihilation group T ∗a by the intertwining operator Φ. This can
be written as M†a = ΦT ∗
a Φ∗. Thus, the F-transform serves as an intertwining
Vol. 72 (2017) Paley-Wiener Isomorphism 2113
operator for the groups M†a on H2
χ. Moreover, using (15), (16) and (17), weobtain
dmTtaf(x)/dtm|t=0 =⟨ε(x) | δ∗m
a Φ∗f⟩w
= dma f(x).
As a result, we have proved the following statement.
Theorem 3. For every f ∈ H2χ the following equalities hold,
TaF(f) = F(M†a f), M†
a f = ΦT ∗a Φ∗f, a ∈ E,
Moreover, for every f ∈ D(φma ) (m ∈ N) and a nonzero a ∈ E,
dma f(x) = 〈ε(x) | δ∗m
a Φ∗f〉w =∫
exp(φx)φma f dχ, x ∈ E.
Let us consider on H2w the multiplicative group with a nonzero a ∈ E,
Ma∗ f(x) = f(x) exp〈x | a〉, f ∈ H2w.
The generator on H2w of the appropriate 1-parameter subgroup C t → Mta∗
is
dMta∗/dt|t=0 = 〈· | a〉 := a∗, a ∈ E.
Hence, it coincides with the following linear operator of multiplication
(a∗f)(x) = 〈x | a〉 f(x) with domain D(a∗) ={f ∈ H2
w : a∗f ∈ H2w
}.
Its power a∗m is densely-defined on D(a∗m) ={f ∈ H2
w : a∗mf ∈ H2w
}which
contains all polynomials from H2w.
Using Lemma 3 we can represent the additive group (E,+) over the spaceH2
χ by the shift group
T †a = ΦTaΦ
∗ with the generator δ†a = ΦδaΦ
∗
defined on D(δ†a) =
{f ∈ H2
χ : δ†af ∈ H2
χ
}This means that T †
a is connected viathe intertwining operator Φ with the exponential creation group Ta.
Theorem 4. For every f ∈ H2χ the following equality holds,
Ma∗F(f) = F(T †a f), a ∈ E,
that is, the F-transform is an intertwining operator for the groups Ma∗ onH2
w and T †a on H2
χ. Moreover, for every f ∈ D(δ†ma ) =
{f ∈ H2
χ : δ†ma f ∈ H2
χ
}
(m ∈ N) and a nonzero a ∈ E,
(a∗mf)(x) = 〈ε(x) | δma Φ∗f〉w =
∫
exp(φx) δ†ma f dχ, x ∈ E. (18)
Proof. The equality (13) yields 〈x | a〉mψ∗n−m(x) =
⟨δ∗ma,nx⊗n | ψn−m
⟩w
forall n ≥ m. By Theorem 2 for any f =
∑n fn ∈ H2
χ there exists a unique
2114 O. Lopushansky Results Math
ψ =⊕
n ψn in Γw with ψn ∈ E�n such that Φ∗f = ψ and fn = ψ∗n. Summing
over all m ∈ Z+ and n ≥ m and using (14), we obtain that
Ma∗ f(x) = exp〈x | a〉⟨ε(x) | Φ∗f
⟩w
=∑
m∈Z+
〈x | a〉m
m!
∑
n≥m
ψ∗n−m(x)
=⟨T ∗a ε(x) | Φ∗f
⟩w
=⟨ε(x) | TaΦ
∗f⟩w.
By Theorem 2 and Lemma 3 it follows that the equalities
Mta∗ f(x) = 〈ε(x) | TtaΦ∗f〉w =
∫
exp(φx)T †taf dχ, t ∈ C (19)
hold for all f ∈ H2w. On the other hand, the equalities (14) and (19) yield
dmMta∗ f(x)dtm
∣∣∣t=0
=∫
exp(φx)dmT †
ta
dtm
∣∣∣t=0
f dχ =∫
exp(φx)δ†ma f dχ
for all f ∈ D(δ†ma ). This in turn yields (18). �
7. Commutation Relations
Describe the commutation relations between M†a and T †
b on the Hardy spaceH2
χ.
Theorem 5. For any nonzero a, b ∈ E the commutation relations
M†aT †
b = exp〈a | b〉T †b M†
a , (φaδ†b − δ†
bφa)f = 〈a | b〉f
hold, wherein f belongs to the dense subspace D(φ2b) ∩ D(δ†2
a ) ⊂ H2χ.
Proof. Let us prove that the following equalities hold,
TaMb∗ = exp〈a | b〉Mb∗Ta, (dab∗ − b∗da)f = 〈a | b〉f (20)
where f ∈ D(b∗2) ∩D(d2a). First property follows from the direct calculations:
Mb∗Taf(x) = exp〈x | b〉f(x + a),
TaMb∗ f(x) = f(x + a) exp〈x | b〉 exp〈a | b〉 = exp〈a | b〉Mb∗Taf(x)
for all f ∈ H2w and x ∈ E. For any f ∈ D(b∗2) ∩ D(d2a) and t ∈ C, we have
d2
dt2TtaMtb∗ f
∣∣t=0
=[d2aTtaMtb∗ f + 2daTtab
∗Mtb∗ f + Ttab∗2Mtb∗ f
]t=0
= (d2a + 2dab∗ + b∗2)f .
Vol. 72 (2017) Paley-Wiener Isomorphism 2115
On the other hand, differentiating again, we have
d
dtTtaMtb∗ f
∣∣t=0
=[ d
dtexp〈ta | tb〉Mtb∗Ttaf + exp〈ta | tb〉 d
dtMtb∗Ttaf
]
t=0,
(d2a + 2dab∗ + b∗2)f =d
dt
[ d
dtTtaMtb∗ f
]
t=0=[ d2
dt2exp〈ta | tb〉Mtb∗Ttaf
+ 2d
dtexp〈ta | tb〉 d
dtMtb∗Ttaf
+ exp〈ta | tb〉 d2
dt2Mtb∗Ttaf
]
t=0
= 2〈a | b〉f + (d2a + 2b∗da + b∗2)f .
This yields (20) where D(b∗2) ∩ D(d2a) contains the dense subspace in H2w of
all polynomials f generating by finite sums Φ∗(f) =⊕
n ψn ∈ Γw.From Mb∗ f(x) = 〈ε(x) | TbΦ
∗f〉w it follows that ITb = Ma∗ I with I :=FΦ. Thus, T †
b = ΦTbΦ∗ = ΦI−1Mb∗ IΦ∗ = F−1Mb∗F . Using that M†
a =F−1TaF with F−1 : H2
w → H2χ and applying (20), we obtain
M†aT †
b = F−1TaMb∗F = exp〈x | b〉F−1Mb∗TaF = exp〈x | b〉T †b M†
a ,
(φaδ†b − δ†
bφa)f = F−1(dab∗ − b∗da)Ff = 〈a | b〉f
for all f ∈ D(φ2b) ∩ D(δ†2
a ). For any f =∑
n fn ∈ H2χ there exists a unique
ψ =⊕
n ψn in Γw with ψn ∈ E�nw such that the equalities Φ∗f = ψ and fn =
ψ∗n hold. Hence, the following embedding D(φ2
b) ∩ D(δ†2a ) ⊂ H2
χ is dense. �
8. Gauss-Weierstrass Semigroups
Next we show that the 1-parameter Gauss-Weierstrass semigroups on theHardy space H2
χ can be well described by shift and multiplicative groups (aclassic case can be found in [7, n.4.3.2]). For this purpose we use the Gaussiankernel
gr(τ) =1√4πr
exp(−τ2
4r
), τ ∈ R, r > 0.
Theorem 6. The 1-parameter Gauss-Weierstrass semigroups{W
δ†a
r : r > 0}
and{W φa
r : r > 0}, defined on the Hardy space H2
χ for any nonzero a ∈ Eas
Wδ†a
r f =∫
R
gr(τ)T †τaf dτ and W φa
r f =∫
R
gr(τ)M†τaf dτ, f ∈ H2
χ, (21)
are generated by δ†2a and φ2
a, respectively.
2116 O. Lopushansky Results Math
Proof. First it is sufficient to prove that the axillary 1-parameter families oflinear operators over H2
w
Ga∗r f =
∫
R
gr(τ)Mτa∗ f dτ and G∂ar f =
∫
R
gr(τ)Tτaf dτ, f ∈ H2w (22)
can be generated by a∗2 and d2a and satisfy the semigroup property. Propertiesof Gaussian kernel yield∫
R
gr(τ)τ2k dτ =1
2√
πr
∫
R
e− τ24r τ2kdτ
∣∣∣τ=2
√rυ
=(2
√r)2k
√π
∫
R
e−υ2υ2k dυ
=22krk
√π
Γ(
2k + 12
)
=2(2k − 1)!(k − 1)!
rk, k ∈ N.
We can rewrite Ga∗r f on the dense subspace
{f ∈ H2
w : exp(τa∗)f ∈ H2w
}as
Ga∗r f =
∫
R
gr(τ) exp(τa∗)f dτ =∑
l∈Z+
a∗lf
l!
∫
R
gr(τ)τ l dτ
=∑
k∈Z+
2(2k − 1)!(k − 1)!
rka∗2kf
(2k)!=
∑
k∈Z+
rka∗2kf
k!= exp(ra∗2)f
By first equality in (22) the family Ga∗r can be extended to the convolution
gr � f :=∫
R
gr(τ)Mτa∗ f dτ, f ∈ H2w
(dependent on a) over the whole space H2w. Thus, to show that the semigroup
property holds, it suffices to show that
gr+s � f = Ga∗r+sf = (Ga∗
r ◦ Ga∗s )f = gr � (gs � f) = (gr ∗ gs) � f .
But this straightly follows from the known convolution equality gr+s = gr ∗gs.Further, using the equality T †
a = F−1Ma∗F we obtain that
Wδ†a
r f =∫
R
gr(τ)F−1Mτa∗Ff dτ = F−1Ga∗r Ff
for all f ∈ H2χ. By Theorem 4 it follows that
dWδ†a
r f
dr
∣∣∣r=0
= F−1Ga∗r f
dr
∣∣∣r=0
= F−1a∗2f = δ†2a f
for all f ∈ D(δ†2a ), since f ∈ D(a∗2) and δ†2
a = F−1a∗2F . Hence, the case of
semigroup Wδ†a
r is proven.Similar reasonings can be applied to the semigroup G∂a
r . As a result, weobtain that the equalities W φa
r = F−1G∂ar F and φ2
a = F−1d2aF hold. �
Vol. 72 (2017) Paley-Wiener Isomorphism 2117
9. Complexified Infinite-Dimensional Heisenberg Group
Let us give yet another application. Consider an infinite-dimensional analogof the Heisenberg group over C. Namely, let us define the group G of uppertriangular matrix-type elements
X(a, b, t) =
⎡
⎣1 a t0 1 b0 0 1
⎤
⎦ , t ∈ C, a, b ∈ E
with unit X(0, 0, 0) and multiplication⎡
⎣1 a t0 1 b0 0 1
⎤
⎦
⎡
⎣1 a′ t′
0 1 b′
0 0 1
⎤
⎦ =
⎡
⎣1 a + a′ t + t′ + 〈a | b′〉0 1 b + b′
0 0 1
⎤
⎦ .
Obviously, X(a, b, t)−1 = X(−a,−b,−t + 〈a | b〉).We will now describe an irreducible linear representation of the group G.
For this purpose we will use the algebra H of quaternions γ = α1 + α2i + β1j+ β2k = (α1 + α2i) + (β1 + β2i)j = α + βj as pairs of complex numbers(α, β) ∈ C
2 with α = α1 + α2i, β = β1 + β2i ∈ C and αı, βı ∈ R (ı = 1, 2)where basis elements in R
4 satisfy the relations i2 = j2 = k2 = ijk = −1,k = ij = −ji, ki = ik = j. Thus, H = C ⊕ Cj is a vector space over C [25].Denote β := �γ where γ = α + βj.
Let EH = E ⊕ Ej be the Hilbert space with H-valued scalar product
〈p | p′〉 = 〈a + bj | a′ + b′j〉 = 〈a | a′〉 + 〈b | b′〉 + [〈a′ | b〉 − 〈a | b′〉] jwhere p = a + bj with a, b ∈ E (similarly, for p′ = a′ + b′j). Hence,
�〈p | p′〉 = 〈a′ | b〉 − 〈a | b′〉, �〈p | p〉 = 0.
The following theorem describes a representation of the above infinite-dimensional Heisenberg group which can be seen as an analog of the Weyl–Schrodinger representation
Theorem 7. The linear representation of G over H2χ
W † : G X(a, b, t) −→ exp[t +
12〈a | b〉
]T †a M†
b
is well defined and irreducible.
Proof. First we prove that the following operator representation
W : G X(a, b, t) −→ exp[t +
12〈a | b〉
]Ma∗Tb
into the algebra of all bounded linear operator on H2w is well defined and
irreducible. Consider the auxiliary group C × EH with the multiplication
(t, p)(t′, p′) =(t + t′ − 1
2�〈p | p′〉, p + p′
)
2118 O. Lopushansky Results Math
for all p = a + bj, p′ = a′ + b′j ∈ EH. It is related to G via the mapping
G : X(a, b, t) −→(t − 1
2〈a | b〉, a + bj
).
Check that G is a group isomorphism. In fact,
G (X(a, b, t)X(a′, b′, t′)) = G (X(a + a′, b + b′, t + t′ + 〈a | b′〉))
=(t + t′ + 〈a | b′〉 − 1
2[〈a + a′ | b + b′〉] , (a + a′) + (b + b′)j
)
=(t + t′ − 1
2[〈a | b〉 + 〈a′ | b′〉
]+
12[〈a | b′〉 − 〈a′ | b〉
], (a + a) + (b + b′)j
)
=(t − 1
2〈a | b〉, a + bj
)(t′ − 1
2〈a′ | b′〉, a′ + b′j
)
= G (X(a, b, t))G (X(a′, b′, t′)) .
Now let us check that the Weyl-like operator
W (p) = exp[12〈a | b〉
]Ma∗Tb, p = a + bj
on the space H2w satisfies the commutation relation
W (p + p′) = exp[
− 12�〈p | p′〉
]W (p)W (p′).
In fact, using (20), we obtain
exp[12〈a | b′〉 − 1
2〈a′ | b〉
]W (p)W (p′)
= exp[12〈a | b〉 +
12〈a′ | b′〉
]exp
[12〈a | b′〉 − 1
2〈a′ | b〉
]Ma∗TbMa′∗Tb′
= exp[12〈a + a′ | b + b′〉
]Ma∗+a′∗Tb+b′ = W (p + p′).
As a consequence, the mapping I : C × EH (t, p) −→ exp(t)W (p) is a groupisomorphism. So, W is also a group isomorphism as a composition of the groupisomorphisms I and G .
Let us check irreducibility. If there exists an element x0 =/ 0 in E and aninteger n > 0 such that
exp[t +
12〈a | b〉
]e〈a|x〉 [x∗
0(x + b)]n = 0 for all x, a, b ∈ E
then x0 = 0. This gives a contradiction. Hence the representation W is irre-ducible. Finally, using that
exp[t +
12〈a | b〉
]T †a M†
b = F−1(
exp[t +
12〈a | b〉
]Ma∗Tb
)F ,
we conclude that the group representation W † = F−1W F is irreducible. �
Vol. 72 (2017) Paley-Wiener Isomorphism 2119
Acknowledgements
I am grateful to Referee for valuable suggestions which improved this article.
Open Access. This article is distributed under the terms of the Creative Com-mons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in anymedium, provided you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate if changeswere made.
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Oleh LopushanskyFaculty of Mathematics and Natural SciencesUniversity of Rzeszow1 Pigonia Str.35-310 RzeszowPolande-mail: [email protected];
Received: April 2, 2017.
Accepted: September 7, 2017.