Weighted Vector-Valued Holomorphic Functions on Banach Spaces

Hindawi Publishing Corporation

Abstract and Applied AnalysisVolume 2013 (2013), Article ID 501592, 9 pageshttp://dx.doi.org/10.1155/2013/501592

Research ArticleWeighted Vector-Valued Holomorphic Functions on Banach SpacesEnrique Jord

Escuela Politcnica Superior de Alcoy, IUMPA, Universitat Politcnica de Valncia, Plaza Ferrndiz y Carbonell 1,03801 Alcoy, Spain

Received 11 February 2013; Accepted 14 May 2013

Academic Editor: Anna Mercaldo

Copyright 2013 Enrique Jord. is is an open access article distributed under the Creative Commons Attribution License, which permitsunrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the weighted Banach spaces of vector-valued holomorphic functions de ned on an open and connected subset of a Banach space. Weuse linearization results on these spaces to get conditions which ensure that a function

1. Introduction, Notation, and Preliminaries

Let

Our main aim here is to analyze a weak criterion for holomorphy and to give extension results for the Banach spaces of holomorphic functionsde ned on a nonvoid open subset

Our notation for the Banach spaces, locally convex spaces, and functional analysis is standard. We refer the reader to [1517]. For a locallyconvex space . We

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de ned in a subset of an open and connected subset of a Banach space , with values in another Banach space , and admitting certain weak extensions in a Banach space of holomorphic

functions can be holomorphically extended in the corresponding Banach space of vector-valued functions.

be a locally convex space. e problem of deciding when a function is holomorphic whenever for each goes back to Dunford [1], who proved that this happens when

quasicomplete. Bogdanowicz [3 is a Banach space. Grothendieck [2] extended the result for

] gives extension results through weak extension, that is, he proved between other results that if two domains (open and connected subsets), satis es that

being are

is a complex, sequentially complete, and locally convex Hausdor space, and admits holomorphic extension for each , then admits a holomorphic extension to . More recently Grosse-

Erdmann, Arendt and Nikolski, Bonet, Frerick, Wengenroth, and the author have given results in this way smoothing the conditions on also requiring extensions of (cf. [48

and]). Also, Laitila and Tylli have recently discussed the dierence only for a proper subset

between strong and weak de nitions for important spaces of vector-valued functions [9, Section 6].

of a Banach space . To obtain these extension results, we use linearization results, that is, theorems whichpermit to identify classes of vector valued functions de ned in and with values in

and with values in with continuous linear mappings from a certain space . Recent work of Beltrn [10], Carando and Zalduendo [11], and Mujica [12] is devoted to get linearization results. We use

for our extension results also linearization results obtained by Bierstedt in [13, 14].

which is nonnormed, we denote by its topological dual. For a Banach space mainly deal with Banach spaces. e absolutely convex hull of a subset closure is taken with respect to other topology and the weak* (

, the dual of is denoted by of is denoted by , and the closure of is denoted by

endowed with the weak (. If the

, it will be denoted by . and are and )) is said to be total if) topology, respectively. e open unit ball of will be denoted by . A subset

is equivalent to (

is ( ) dense. By the Hahn Banach theorem, being total in is said to be norming if

being separating, that is, if and for all , then . is bounded, and its associated functional ,

de nes an equivalent norm in (closed) unit ball in subset

, that is, if the polar set de nes an equivalent. It is immediate that if is norming then it is also separating. If is the norm of , then is called 1-norming. A

is called norming or total when we consider determine boundedness whenever all the is a norming subset of

, that is, de nes an equivalent norm in . A subspace of is said to be

is said tonorming if -bounded subsets of are ( -) bounded. A subspace

. We give below a relation between these concepts. e result is given in [6, Proposition 7, Remark 8] in the moregeneral context of the Frchet spaces, though in this paper the norming subspaces are called almost norming. We give a proof for the Banachcase because it is very transparent.

Weighted Vector-Valued Holomorphic Functions on Banach Spaces http://www.hindawi.com/journals/aaa/2013/501592/

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Proposition 1 (see [6]). Let is norming if and only if

Proof. It is standard to check that is implies that . By the very de nition, of to Boundedness Principle, the bounded since the norms are supposed to beequivalent in

us, the property of being norming for subspaces in is between weak*-dense and strongly dense.

Let

equipped with the norm which makes it isomorphic to a quotient of known.

Lemma 2. Let

Proof. e hypothesis on

Hence, we take polars and apply the bipolar theorem to get

Let

We get from the Schauder lemma [16, Lemma 3.9] that

Remark 3. If we assume in Lemma 2 that

We see below that if the bounded subset

Remark 4 (Bonet). e assertion in Lemma 2 implies that if and

(see [17

From

Let

Recall that a function that

Analogously, for a Banach space

be a Banach space. A subspace determines boundedness in .

on . From this, it follows that is norming if and only if is. Assume rst that is norming. is separating on , and then, we can consider the algebraic inclusion is the restriction

. e hypothesis norming means that is isomorphic to , which is a subspace of , and then,

. By the Uniform bounded subsets of are norm bounded in

. Conversely, let one assume that that the identity implies that

determines boundedness in . is implies, again by the Uniform Boundedness Principle, is bounded. Hence, there exists such that for each , which

is norming.

be a bounded subset of and an index set. Let

. We will use the following lemma, which we supposed to be well

be a norming subset. en, the injection of in is an onto isomorphism.

yields that there exist such that

be the equivalent open unit ball in such that . We de ne , . is clearly bounded. Moreover,, and then,

is open and then surjective. We conclude from the very de nition of .

is 1-norming, then the isomorphism is an isometry.

is not norming, then the assertion is not true in general.

is bounded, then if and only if for each there exist sequences such that . is is not in general true if we assume that is only to be bounded. Valdivia showed

, Example 3.2.21]) that in every in nite dimensional Banach space there is an absolutely convex bounded subset closed such that

which is not is a Banach space with closed unit ball , where

being Banach, we conclude , but from the proof given in [17, Example ], it follows that is not included in .

be a connected open subset of a Banach space topology on called If

. e space of all the holomorphic functions on is denoted by . e compact open denoted by . is a semi-Montel space; that is, each closed and bounded subset is compact. A subset is

-bounded if it is bounded and the distance of to the complementary of is positive. For , -bounded means simply bounded.. If is the unit ball of a Banach space,

-valued holomorphic functions on is -bounded if and only if it is contained in a ball of radius is a Banach space, the space of

is denoted by . We refer to [18] for the precise de nitions. A weight continuous function which is strictly positive. According to [19], we say that a weight below in each

is a on satis es the property (I) whenever it is bounded

-bounded subset of . e weighted Banach spaces of holomorphic functions are de ned as

is said to vanish at in nity on -bounded sets when for each there exists a -bounded subset such for

the space of . is always continuously embedded in

holomorphic functions of bounded type -bounded sets.

. If satis es , then is continuously embedded in, which is endowed with the Frchet topology of uniform convergence on

, we de ne the weighted spaces of vector-valued functions as


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During all the work, our model spaces will be corresponding vector-valued analogues they are the spaces

Let

In case If

e sampling sets (as well as interpolation sets) of the weighted space in [23] in terms of certain densities.

2. Banach Subspaces of

Let one consider

Since weakly holomorphic functions are holomorphic and weakly bounded sets are bounded, it follows that for agrees with the strong de nition given previously. Following the same steps as in [12, eorem 2.1] (also in [

Remark 5. In [25], it is shown that such that denote uniqueness for

Proposition 6. Let that

Proof. If

Conversely, let

and then,

Now, we are going to show that there are more natural spaces with compact unit ball for the compact open topology. To do this, we present ageneral result of complemented subspaces in the Frchet spaces of analytic functions which could be of independent interest. We state it forFrchet instead of Banach to include the important space in [19, Proposition 3, Example 14].

eorem 7. Let then

Proof. Let one assume without loss of generality that

Let

and . But we will deal with general closed subspaces of and their (which will be de ned in the following section) in order to consider important subspaces as

of homogeneous polynomials of degree and, in case of holomorphic and bounded functions which are continuous and uniformly continuous on

being bounded, the algebras and of, respectively.

be a subspace of null. A set

. A subset is said to be a set of uniqueness for if each which vanishes at such that, for every

is identically is said to be sampling for if there exists some constant ,

is an algebra the constant, can be always taken 1 and, according to Globevnik, the sampling sets are called boundaries [2022]., it follows from the de nitions that , and is sampling if and only if is norming, and

is a set of uniqueness if and only if is total.

(i.e., for , ) were characterized by Seip

Which Are Dual Spaces

to be a subspace with compact closed unit ball for closed. We de ne the Banach space of vector-valued functions in a weak sense:

. Notice that this condition implies that is norm

this de nition24, Lemma 10]), we get that

can be identi ed with , being the predual of that exists by the Dixmier-Ng eorem [25].

consists of all the functionals , we have that

restricted to is continuous. Let . If we is separating in if and only if is (weakly) dense if and only if is a set of

. Analogously, is norming in if and only if is sampling for .

be a subspace of with compact closed unit ball. en, if and only if there exists such. Moreover, the correspondence is an isometry.

, we de ne . Since is continuous and , it follows that for each , and then, by the very de nition.

. We set . Since is 1-norming in , we apply Lemma 2 and Remark 3, to get that. We see that

, de nes a linear mapping on . If , then, for each

is well de ned. Moreover, since , it is easy to compute .

. For and with radial and balanced, it is done by Garca et al.

be a Frchet space of holomorphic functions on such that continuously. If, for , , endowed with its norm topology is a complemented subspace of . If , then is compact in .

. For , we denote the -homogeneous polynomial such that

be denoted by the topology in of pointwise convergence on . e projection


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is continuous. We checked it. Let and let

Hence, by the closed graph theorem, we get that the map surjective and restricted to

We check now that convergent to

For spaces

Proposition 8. Let

Proof. Let

From the linearization of these dual Banach subspaces of functions [4, eorem 2.5] generalized in [7

Proposition 9. Let Banach space. If

Proof. Let

By hypothesis, there exists

us,

for each on

Proposition 9 and eorem 7 yield that the Banach-Steinhaus theorem stated as in [27, 39.4(1)] can be extended to the space of vector-valuedpolynomials

e following results are extensions of those obtained in [7] by Frerick et al. for spaces of bounded holomorphic and harmonic functions onopen subsets of nite-dimensional subspaces with values in locally convex spaces. Our results are valid for spaces of functions de ned on anopen and connected subset

, and let be a net convergent to in . Let such that the closed ball , with . For , we de ne , being the ball with radius in . Let . We have that

converges to in . We conclude from the continuity of the evaluations of the derivatives in this last space and

, is continuous. Since the map is by hypothesis is an that is the identity, it also follows that

isomorphism. Hence, the inverse of the inclusion 2, Section 7, Proposition 3] to conclude that

is closed in . us, the inclusion satis es that is the identity in . We apply [26, Chapter

is complemented in .

is compact for the topology of pointwise convergence on . Let be a net in such that it is pointwise in . Assume without loss of generality that is nonempty. e net is a bounded net in

which is Cauchy for the topology of pointwise convergence in convergence in

. is topology is Hausdor and weaker than the topology of pointwise. Since is a dual space [18, Proposition 1.17], the topology of pointwise convergence on

compact restricted to the bounded sets in Moreover, get

is relatively and then agrees in the bounded sets with the topology of pointwise convergence on .

is bounded in , and hence, we get that is convergent to pointwise in . Since , we. We have proved that is closed in for the topology of pointwise convergence in , and then it is compact.

containing with its natural norm

, we have that is a subspace which is complemented and it is isomorphic to endowed. Moreover, has a compact unit ball for the topology of pointwise convergence in

dual Banach space because of Dixmier-Ng theorem [25]. We denote by , and hence, it is a

the predual of and by the predual of obtained in [12, eorem 2.4]. In

subspace. e same applies for independent by [11

, the subset is norming and then spans a ( -) dense in . Both and are formed by functionals which are linearly

, Proposition 1]. We check below that there is a natural isomorphism between and .

be a weight on such that . e predual of is isomorphic to the predual of canonically, that is, there exists such that for each .

be as de ned previously. If we de ne , by means of , we have that is well de ned since, it is (weakly) continuous, and then, it can be extended to

subspace of . If we consider now as a

it is (weakly) dense since open sets are sets of uniqueness in (weakly) continuous, and hence that we get, an extension the identity since both coincide in Section 7, Proposition 3].

. e linear map , is again. is a continuous linear mapping, and then, it is

is an onto isomorphism by [26, Chapter 2,. Moreover, has dense range in . Hence,

, one can get easily an extension of the Blaschke-type result for vector-valued, Corollary 4.2]. e proof that we give is strongly based on the Banach-Steinhaus principle.

be a subspace of which has a -compact closed unit ball, let be a set of uniqueness for , and let be a is a bounded net in such that is convergent for each , then is convergent to a function

uniformly on the compact subsets of .

be the sequence of operators in 1-norming subset of

such that for each . Let . is a, that is,

such that

. By Remark 5, the subset is total in . Since is equicontinuous, the topology of pointwise convergence coincides with the topology of pointwise convergence in

convergence is uniform on the compact subsets of from the observation that

by [27, 39.4(1)]. us, is pointwise convergent to . e by [27, 39.4(2)]. If is compact, then is compact in . is follows

, is (weakly) holomorphic and then continuous.

. Bochnak and Siciak showed [28, eorem 2] that the uniform boundedness principle also is valid for polynomials.

of a Banach space , but we restrict to the case of Banach-valued functions. e proofs that we give here are


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simpler. e next theorem extends [7, eorem 2.2].

eorem 10. Let

Proof. Let

compute:

Since this is true for each bounded linear mapping. Since

e following result is a generalization of [6, eorem 1(ii)].

eorem 11. Let

Proof. If Proposition 9Proposition

We now study the problem of extending functions which admit extensions for functionals in a subspace which we assume only to be

eorem 12. Let

Proof. e set

e open unit ball

Since

Remark 13. If we consider , . Moreover, itis immediate that

3. General Banach Subspaces of

For arbitrary Banach spaces

A Banach space

be a weight on be a Banach space, and let extension

, let be a subspace of with -compact closed unit ball, let be a subspace which determines boundedness in

be a set of uniqueness for , let . If is a function such that admits an

for each , then admits a unique extension .

be the span of . e hypothesis implies that is dense, and then, it is dense in norm. e map be an element in the unit ball of , is well de ned since is separating. Let , and let . We

, we conclude that is bounded and then norm bounded by hypothesis. us, is a is dense in , we can extend to . We conclude by Proposition 6.

be a weight on be a Banach space, and let

, let be a subspace of with -compact closed unit ball, let be a set of uniqueness for , let be a norming subspace. If is a function such that admits an extension for each

such that is bounded in , then admits a unique extension .

and tend to yields that there exists

1 and eorem 10.

, then is a bounded sequence such that converges to for each . such that tends to for each . e conclusion is a consequence of

of dense. In this case, we require that

and Bonet et al. [5 is quite large. is is symmetric with the problem studied by Gramsch [29], Grosse-Erdmann [8],

]. e next theorem is an extension to our context of [7, eorem 3.2].

be a weight on Banach space, and let extension

, let be a subspace of . If

with -compact unit ball, and let be a sampling set for . Let be a be a -dense subspace of is a function such and such that admits an

for each , then there exists a unique extension of .

is norming for means that for each

; hence, we apply Lemma 2 to get that is isomorphic to . is, there exists and such that

in for the norm which makes this space isometric to is formed by the vectors . We de ne that

such that the sequence in the previous representation can be taken in the open unit ball of , . Since

is bounded by hypothesis, the series is convergent. Moreover, if , then for each

is separating, is well de ned. Moreover, the hypothesis of boundedness of we conclude by Proposition 6.

implies that is bounded. Hence,

, we have that ; hence, is relatively compact in admits an extension to for each (the space of sequences which are zero but nitely many components), is

dense (even norming since it is dense in ), and is a set of uniqueness for . However, since for each . is shows that the hypothesis in eorems 10 and 12 is optimal, that is, for the conditions on the set

where the functions are de ned and in the subspace for which functionals, we have weak extensions that cannot be simultaneously relaxed,and also the condition of boundedness in the extensions in eorem 11 can not be dropped.

with no assumption on the unit ball, the equivalence between the weak and the strong de nitionsdoes not hold in general. We discuss it below. We consider the space , and we de ne

is said to satisfy the Schur property if every sequence well-known theorem of Schur asserts that satis es this property.

in which is weakly convergent is also norm convergent. e


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Proposition 14. If

Proof. Suppose that there exists

is (weakly) convergent to zero, a contradiction.

We see below that the situation diers for function with values in the general Banach spaces.

Example 15. Assume that

Proof. First, we proceed similarly as in [30, Lemma 21] to get a sequence such that such that that

We consider be arbitrary. Since

Since

us,

Example 16. Assume that

Proof. e hypothesis on

then for each Cauchy in

De ning , we have that

e proof is complete since not re exive [32, 33

such that

Since

hence,

us, on the contrary that with the concrete examples of dual spaces

In view of Proposition 14, one could expect that the analogous extensions of eorems 10 and 12 are possible for to have the Schur property. is is not the case as the following example shows.

Example 17. Let with .Consider that

(a)

is a Banach space with the Schur property, then .

. en, there exist . is last condition implies that

and going to in nity on -bounded sets such that and for all

is nite dimensional and is in nite dimensional. en, .

in converges to 0 in such that

, and there exists for all . Since is in nite dimensional, there is and for . We apply

is metrizable and is compact to get that is relatively sequentially compact. Hence, we can extract a subsequence of which is Cauchy for , and we denote again by . De ning , we get the desired sequence.

, . Let , the series is convergent in . Hence,. e convergence of for the compact open topology implies that for each there exists such that

, we obtain that there exists such that

.

is the unit ball of a Banach space , continuous with for and and for . en, .

implies that for each consider the Cesro means

the Taylor polynomials of the development at zero converge to in . If we

([31, Proposition 1.2], [19, Proposition 4]) and in . If , then is not, since is closed. Hence, there are and a subsequence such that

tends to 0 in . Proceeding as in Example 15, we obtain that satis es.

is never empty. We checked it. If ]. Hence, there exists

and

, then is the bidual of and this last space is with

. De ne . For arbitrary, we consider such that and

by . Since is nonincreasing, we have

, there exist and a sequence of complex numbers smaller than 1 such that and

.

considered in the previous section ( and ), in the de nition of the corresponding spaces of vector-valued functions in the weak sense are not consistent with the natural de nition. Forlinearization for these spaces with the weak de nition, we refer to the work of Carando and Zalduendo [11].

when is required

be the unit ball of a Banach space , and let for . Fix with and , , then the following applies.

and for each


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Proof. To prove (a), we observe that

Let

Let

From (30), (31), and

Remark 18. e same computation as in Example 17(b) shows that for

4. Spaces of Weighted Compact Range Vector-Valued Holomorphic Functions

In this section, we consider the natural extension to the weighted case of the vector-valued compact holomorphic functions introduced by Aronand Schottenloher in [34] by means of the weak de nition, that is, for an open and connected subset

In case

We check that this (weak) de nition agrees with the natural de nition when holomorphic and uniformly continuous functions on

Assume that . Given

Let one assume that

Given two locally convex spaces

(b)Hence,

is increasing for ; hence,

. Let and such that implies . For each , since , we have

such that for each

, we obtain that

and the function , satis es that .

of a Banach space , a closed subspace of , and a Banach space , we de ne that

is nite dimensional, the space compacti cation

is the space of holomorphic functions such that is continuous in the Alexandro of and . Hence, in this case. If is in nite dimensional, the inclusion

is strict in general. Observe that if is the unit ball and vanishes at on , then .

is the unit ball of , , and the space of the, that is, we want to show that

satis es that neighbourhood

for each , since is relatively compact, there exists a weak of 0 such that

. Since is uniformly continuous on for , there exists such that implies , and therefore, .

and , we denote by its -product of Schwartz, that is, the space of all linear and continuous mappings is , endowed with the topology of uniform convergence on the equicontinuous subsets of

of uniform convergence on the convex compact subsets of . endowed with the topology

. e -product is symmetric by means of the transpose mapping [27, 43.3(3)]. In

. for each .


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case

eorem 19. Let

Proof. If -

If (ii) is satis ed, then the transpose to a relatively compactsubset of

Finally, that (iii) implies (i) is trivial.

Observe that, setting

We nish showing that the weak de nition given in this section for

We use a similar argument to the one used by Bierstedt in [13, page 200] in a more general setting, including our case when dimensional (i.e., putting compact instead of , then by the previous theorem, there exists

Acknowledgments

e author wants to thank J. Bonet for several references, discussions, and ideas provided, which were very helpful and in particular allowedhim to prove eorem 7, Proposition 8, and Examples 15 and 16. Remark 4 is due to him. e participation of M. J. Beltrn in a lot ofdiscussions during all the work has also been very important. Her ideas are also re ected in the paper. e author is also indebted to L. Frerickand J. Wengenroth for communicating to him Lemma 2. e remarks and corrections of the referee have been also really helpful to the nalversion. e author thanks him/her for that. is research was partially supported by MEC and FEDER Project MTM2010-15200, GV ProjectACOMP/2012/090, and Programa de Apoyo a la Investigacin y Desarrollo de la UPV PAID-06-12.

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Weighted Vector-Valued Holomorphic Functions on Banach Spaces

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