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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 501592, 9 pages
Weighted Vector-Valued Holomorphic Functions on Banach Spaces
Escuela Politécnica Superior de Alcoy, IUMPA, Universitat Politècnica de València, Plaza Ferrándiz y Carbonell 1, 03801 Alcoy, Spain
Received 11 February 2013; Accepted 14 May 2013
Academic Editor: Anna Mercaldo
Copyright © 2013 Enrique Jordá. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the weighted Banach spaces of vector-valued holomorphic functions defined on an open and connected subset of a Banach space. We use linearization results on these spaces to get conditions which ensure that a function defined 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.
1. Introduction, Notation, and Preliminaries
Let be a locally convex space. The problem of deciding when a function is holomorphic whenever for each goes back to Dunford , who proved that this happens when is a Banach space. Grothendieck  extended the result for being quasicomplete. Bogdanowicz  gives extension results through weak extension, that is, he proved between other results that if are two domains (open and connected subsets), is a complex, sequentially complete, and locally convex Hausdorff space, and satisfies that 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 and also requiring extensions of only for a proper subset (cf. [4–8]). Also, Laitila and Tylli have recently discussed the difference between strong and weak definitions for important spaces of vector-valued functions [9, Section 6].
Our main aim here is to analyze a weak criterion for holomorphy and to give extension results for the Banach spaces of holomorphic functions defined on a nonvoid open subset of a Banach space . To obtain these extension results, we use linearization results, that is, theorems which permit to identify classes of vector valued functions defined in and with values in with continuous linear mappings from a certain space and with values in . Recent work of Beltrán , Carando and Zalduendo , and Mujica  is devoted to get linearization results. We use for our extension results also linearization results obtained by Bierstedt in [13, 14].
Our notation for the Banach spaces, locally convex spaces, and functional analysis is standard. We refer the reader to [15–17]. For a locally convex space which is nonnormed, we denote by its topological dual. For a Banach space , the dual of is denoted by . We mainly deal with Banach spaces. The absolutely convex hull of a subset of is denoted by , and the closure of is denoted by . If the closure is taken with respect to other topology , it will be denoted by . and are and endowed with the weak () and the weak* () topology, respectively. The open unit ball of will be denoted by . A subset () is said to be total if is () dense. By the Hahn Banach theorem, being total in is equivalent to being separating, that is, if and for all , then . is said to be norming if is bounded, and its associated functional , defines an equivalent norm in , that is, if the polar set defines an equivalent (closed) unit ball in . It is immediate that if is norming then it is also separating. If is the norm of , then is called 1-norming. A subset is called norming or total when we consider , that is, defines an equivalent norm in . A subspace of is said to determine boundedness whenever all the -bounded subsets of are (-) bounded. A subspace is said to be norming if is a norming subset of . We give below a relation between these concepts. The result is given in [6, Proposition 7, Remark 8] in the more general context of the Fréchet spaces, though in this paper the norming subspaces are called almost norming. We give a proof for the Banach case because it is very transparent.
Proposition 1 (see ). Let be a Banach space. A subspace is norming if and only if determines boundedness in .
Proof. It is standard to check that on . From this, it follows that is norming if and only if is. Assume first that is norming. This implies that is separating on , and then, we can consider the algebraic inclusion . By the very definition, is the restriction of to . The hypothesis norming means that is isomorphic to , which is a subspace of . By the Uniform Boundedness Principle, the bounded subsets of are norm bounded in , and then, bounded since the norms are supposed to be equivalent in . Conversely, let one assume that determines boundedness in . This implies, again by the Uniform Boundedness Principle, that the identity is bounded. Hence, there exists such that for each , which implies that is norming.
Thus, the property of being norming for subspaces in is between weak*-dense and strongly dense.
Let be a bounded subset of and an index set. Let equipped with the norm which makes it isomorphic to a quotient of . We will use the following lemma, which we supposed to be well known.
Lemma 2. Let be a norming subset. Then, the injection of in is an onto isomorphism.
Proof. The hypothesis on yields that there exist such that Hence, we take polars and apply the bipolar theorem to get Let be the equivalent open unit ball in such that . We define , . is clearly bounded. Moreover, , and then, We get from the Schauder lemma [16, Lemma 3.9] that is open and then surjective. We conclude from the very definition of .
Remark 3. If we assume in Lemma 2 that is 1-norming, then the isomorphism is an isometry.
We see below that if the bounded subset is not norming, then the assertion is not true in general.
Remark 4 (Bonet). The assertion in Lemma 2 implies that if is bounded, then if and only if for each there exist sequences and such that . This is not in general true if we assume that is only to be bounded. Valdivia showed (see [17, Example 3.2.21]) that in every infinite dimensional Banach space there is an absolutely convex bounded subset which is not closed such that is a Banach space with closed unit ball , where From being Banach, we conclude , but from the proof given in [17, Example ], it follows that is not included in .
Let be a connected open subset of a Banach space . The space of all the holomorphic functions on is denoted by . The compact open topology on denoted by . is a semi-Montel space; that is, each closed and bounded subset is compact. A subset is called -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, is -bounded if and only if it is contained in a ball of radius . If is a Banach space, the space of -valued holomorphic functions on is denoted by . We refer to  for the precise definitions. A weight is a continuous function which is strictly positive. According to , we say that a weight on satisfies the property (I) whenever it is bounded below in each -bounded subset of . The weighted Banach spaces of holomorphic functions are defined as Recall that a function is said to vanish at infinity on -bounded sets when for each there exists a -bounded subset such that for . is always continuously embedded in . If satisfies , then is continuously embedded in the space of holomorphic functions of bounded type , which is endowed with the Fréchet topology of uniform convergence on -bounded sets.
Analogously, for a Banach space , we define the weighted spaces of vector-valued functions as During all the work, our model spaces will be and . But we will deal with general closed subspaces of and their corresponding vector-valued analogues (which will be defined in the following section) in order to consider important subspaces as they are the spaces of homogeneous polynomials of degree and, in case of being bounded, the algebras and of holomorphic and bounded functions which are continuous and uniformly continuous on , respectively.
Let be a subspace of . A subset is said to be a set of uniqueness for if each which vanishes at is identically null. A set is said to be sampling for if there exists some constant such that, for every ,
In case is an algebra the constant, can be always taken 1 and, according to Globevnik, the sampling sets are called boundaries [20–22]. If and , it follows from the definitions that , is sampling if and only if is norming, and is a set of uniqueness if and only if is total.
The sampling sets (as well as interpolation sets) of the weighted space (i.e., for , ) were characterized by Seip in  in terms of certain densities.
2. Banach Subspaces of Which Are Dual Spaces
Let one consider to be a subspace with compact closed unit ball for . Notice that this condition implies that is norm closed. We define the Banach space of vector-valued functions in a weak sense: Since weakly holomorphic functions are holomorphic and weakly bounded sets are bounded, it follows that for this definition agrees with the strong definition given previously. Following the same steps as in [12, Theorem 2.1] (also in [24, Lemma 10]), we get that can be identified with , being the predual of that exists by the Dixmier-Ng Theorem .
Remark 5. In , it is shown that consists of all the functionals such that restricted to is continuous. Let . If we denote , we have that is separating in if and only if is (weakly) dense if and only if is a set of uniqueness for . Analogously, is norming in if and only if is sampling for .
Proposition 6. Let be a subspace of with compact closed unit ball. Then, if and only if there exists such that . Moreover, the correspondence is an isometry.
Proof. If , we define . Since is continuous and , it follows that for each , and then, by the very definition.
Conversely, let . We set . Since is 1-norming in , we apply Lemma 2 and Remark 3, to get that . We see that defines a linear mapping on . If , then, for each , and then, is well defined. Moreover, since , it is easy to compute .
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 a general result of complemented subspaces in the Fréchet spaces of analytic functions which could be of independent interest. We state it for Fréchet instead of Banach to include the important space . For and with radial and balanced, it is done by García et al. in [19, Proposition 3, Example 14].
Theorem 7. Let be a Fréchet space of holomorphic functions on such that continuously. If, for , , then endowed with its norm topology is a complemented subspace of . If , then is compact in .
Proof. Let one assume without loss of generality that . For , we denote the -homogeneous polynomial such that
Let be denoted by the topology in of pointwise convergence on . The projection
is continuous. We checked it. Let , and let be a net convergent to in . Let such that the closed ball , and let with . For , we define , 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
Hence, by the closed graph theorem, we get that the map , is continuous. Since the map is by hypothesis surjective and restricted to that is the identity, it also follows that is closed in . Thus, the inclusion is an isomorphism. Hence, the inverse of the inclusion satisfies that is the identity in . We apply [26, Chapter 2, Section 7, Proposition 3] to conclude that is complemented in .
We check now that is compact for the topology of pointwise convergence on . Let be a net in such that it is convergent to pointwise in . Assume without loss of generality that is nonempty. The net is a bounded net in which is Cauchy for the topology of pointwise convergence in . This topology is Hausdorff and weaker than the topology of pointwise convergence in . Since is a dual space [18, Proposition 1.17], the topology of pointwise convergence on is relatively compact restricted to the bounded sets in and then agrees in the bounded sets with the topology of pointwise convergence on . Moreover, is bounded in , and hence, we get that is convergent to pointwise in . Since , we get . We have proved that is closed in for the topology of pointwise convergence in , and then it is compact.
For spaces containing , we have that is a subspace which is complemented and it is isomorphic to endowed with its natural norm . Moreover, has a compact unit ball for the topology of pointwise convergence in , and hence, it is a dual Banach space because of Dixmier-Ng theorem . We denote by the predual of and by the predual of obtained in [12, Theorem 2.4]. In , the subset is norming and then spans a (-) dense subspace. The same applies for in . Both and are formed by functionals which are linearly independent by [11, Proposition 1]. We check below that there is a natural isomorphism between and .
Proposition 8. Let be a weight on such that . The predual of is isomorphic to the predual of canonically, that is, there exists such that for each .
Proof. Let be as defined previously. If we define, by means of , we have that is well defined since , it is (weakly) continuous, and then, it can be extended to . If we consider now as a subspace of it is (weakly) dense since open sets are sets of uniqueness in . The linear map , is again (weakly) continuous, and hence that we get, an extension . is a continuous linear mapping, and then, it is the identity since both coincide in . Moreover, has dense range in . Hence, is an onto isomorphism by [26, Chapter 2, Section 7, Proposition 3].
From the linearization of these dual Banach subspaces of , one can get easily an extension of the Blaschke-type result for vector-valued functions [4, Theorem 2.5] generalized in [7, Corollary 4.2]. The proof that we give is strongly based on the Banach-Steinhaus principle.
Proposition 9. Let be a subspace of which has a -compact closed unit ball, let be a set of uniqueness for , and let be a Banach space. If is a bounded net in such that is convergent for each , then is convergent to a function uniformly on the compact subsets of .
Proof. Let be the sequence of operators in such that for each . Let . is a 1-norming subset of , that is, By hypothesis, there exists such that Thus, for each . By Remark 5, the subset is total in . Since is equicontinuous, the topology of pointwise convergence on coincides with the topology of pointwise convergence in by [27, 39.4(1)]. Thus, is pointwise convergent to . The convergence is uniform on the compact subsets of by [27, 39.4(2)]. If is compact, then is compact in . This follows from the observation that , is (weakly) holomorphic and then continuous.
Proposition 9 and Theorem 7 yield that the Banach-Steinhaus theorem stated as in [27, 39.4(1)] can be extended to the space of vector-valued polynomials . Bochnak and Siciak showed [28, Theorem 2] that the uniform boundedness principle also is valid for polynomials.
The following results are extensions of those obtained in  by Frerick et al. for spaces of bounded holomorphic and harmonic functions on open subsets of finite-dimensional subspaces with values in locally convex spaces. Our results are valid for spaces of functions defined on an open and connected subset of a Banach space , but we restrict to the case of Banach-valued functions. The proofs that we give here are simpler. The next theorem extends [7, Theorem 2.2].
Theorem 10. Let be a weight on , let be a subspace of with -compact closed unit ball, let be a set of uniqueness for , let be a Banach space, and let be a subspace which determines boundedness in . If is a function such that admits an extension for each , then admits a unique extension .
Proof. Let be the span of . The hypothesis implies that is dense, and then, it is dense in norm. The map , is well defined since is separating. Let be an element in the unit ball of , and let . We compute:
Since this is true for each , we conclude that is bounded and then norm bounded by hypothesis. Thus, is a bounded linear mapping. Since is dense in , we can extend to . We conclude by Proposition 6.
The following result is a generalization of [6, Theorem 1(ii)].
Theorem 11. Let be a weight on , let be a subspace of with -compact closed unit ball, let be a set of uniqueness for , let be a Banach space, and 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 .
Proof. If and tend to , then is a bounded sequence such that converges to for each . Proposition 9 yields that there exists such that tends to for each . The conclusion is a consequence of Proposition 1 and Theorem 10.
We now study the problem of extending functions which admit extensions for functionals in a subspace of which we assume only to be dense. In this case, we require that is quite large. This is symmetric with the problem studied by Gramsch , Grosse-Erdmann , and Bonet et al. . The next theorem is an extension to our context of [7, Theorem 3.2].
Theorem 12. Let be a weight on , let be a subspace of with -compact unit ball, and let be a sampling set for . Let be a Banach space, and let be a -dense subspace of . If is a function such and such that admits an extension for each , then there exists a unique extension of .
Proof. The set is norming for ; hence, we apply Lemma 2 to get that is isomorphic to . This means that for each , there exists and such that The open unit ball in for the norm which makes this space isometric to is formed by the vectors such that the sequence in the previous representation can be taken in the open unit ball of . We define that , . Since is bounded by hypothesis, the series is convergent. Moreover, if , then for each Since is separating, is well defined. Moreover, the hypothesis of boundedness of implies that is bounded. Hence, we conclude by Proposition 6.
Remark 13. If we consider , , we have that ; hence, is relatively compact in . Moreover, it is immediate that admits an extension to for each (the space of sequences which are zero but finitely many components), is dense (even norming since it is dense in ), and is a set of uniqueness for . However, since for each . This shows that the hypothesis in Theorems 10 and 12 is optimal, that is, for the conditions on the set where the functions are defined 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 Theorem 11 can not be dropped.
3. General Banach Subspaces of
For arbitrary Banach spaces with no assumption on the unit ball, the equivalence between the weak and the strong definitions does not hold in general. We discuss it below. We consider the space , and we define A Banach space is said to satisfy the Schur property if every sequence in which is weakly convergent is also norm convergent. The well-known theorem of Schur asserts that satisfies this property.
Proposition 14. If is a Banach space with the Schur property, then .
Proof. Suppose that there exists . Then, there exist and going to infinity on -bounded sets such that and for all . This last condition implies that is (weakly) convergent to zero, a contradiction.
We see below that the situation differs for function with values in the general Banach spaces.
Example 15. Assume that is finite dimensional and is infinite dimensional. Then, .
Proof. First, we proceed similarly as in [30, Lemma 21] to get a sequence in such that converges to 0 in , and there exists such that for all . Since is infinite dimensional, there is and such that for . We apply that 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 . Defining , we get the desired sequence.
We consider , . Let be arbitrary. Since , the series is convergent in . Hence, . The convergence of for the compact open topology implies that for each there exists such that Since , we obtain that there exists such that Thus, .
Example 16. Assume that is the unit ball of a Banach space , continuous with for and and for . Then, .
Proof. The hypothesis on implies that for each the Taylor polynomials of the development at zero converge to in . If we consider the Cesàro means
then for each ([31, Proposition 1.2], [19, Proposition 4]) and in . If , then is not Cauchy in , since is closed. Hence, there are and a subsequence such that
Defining , we have that tends to 0 in . Proceeding as in Example 15, we obtain that satisfies .
The proof is complete since is never empty. We checked it. If , then is the bidual of and this last space is not reflexive [32, 33]. Hence, there exists with . For arbitrary, we consider such that and such that and . Define by . Since is nonincreasing, we have Since , there exist and a sequence of complex numbers smaller than 1 such that and hence, .
Thus, on the contrary that with the concrete examples of dual spaces considered in the previous section ( and ), in the definition of the corresponding spaces of vector-valued functions in the weak sense are not consistent with the natural definition. For linearization for these spaces with the weak definition, we refer to the work of Carando and Zalduendo .
In view of Proposition 14, one could expect that the analogous extensions of Theorems 10 and 12 are possible for when is required to have the Schur property. This is not the case as the following example shows.
Example 17. Let be the unit ball of a Banach space , and let for . Fix with and with . Consider that , , then the following applies. (a) and for each Hence, . (b) for each .
Remark 18. The same computation as in Example 17(b) shows that for and the function , satisfies that .
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 Aron and Schottenloher in  by means of the weak definition, that is, for an open and connected subset of a Banach space , a closed subspace of , and a Banach space , we define that
In case is finite dimensional, the space is the space of holomorphic functions such that is continuous in the Alexandroff compactification of and . Hence, in this case. If is infinite dimensional, the inclusion is strict in general. Observe that if is the unit ball and vanishes at on , then .
We check that this (weak) definition agrees with the natural definition when is the unit ball of , , and the space of the holomorphic and uniformly continuous functions on , that is, we want to show that
Assume that satisfies that for each . Given , since is relatively compact, there exists a weak neighbourhood of 0 such that Let one assume that . Since is uniformly continuous on for , there exists such that implies , and therefore, .
Given two locally convex spaces and , we denote by its -product of Schwartz, that is, the space of all linear and continuous mappings , endowed with the topology of uniform convergence on the equicontinuous subsets of . is endowed with the topology of uniform convergence on the convex compact subsets of . The -product is symmetric by means of the transpose mapping [27, 43.3(3)]. In case and being Banach spaces, belongs to if and only if is a compact operator which is weak*-weak continuous by [27, 43.3(2)]. The next theorem is the analogous of Theorem 12 in the case of general Banach spaces of functions, not necessarily dual Banach spaces. However, the techniques used here are different. The proof is analogous to the one given by Bierstedt and Holtmanns in  when the linearization result is obtained in a much more general context, but we only require the function to be defined in a sampling set.
Theorem 19. Let be a closed subspace of , and let be a sampling set for . Let be a Banach space, and let be a weak*-dense subspace of . The following are equivalent. (i) satisfies that is relatively compact in , and admits an extension for each . (ii)The linear mapping , admits an extension . (iii) can be extended to .
Proof. If satisfies (i), then the linear mapping , is - continuous, since the absolute convex hull of is relatively compact in , and the uniform convergence on defines an equivalent topology in . By the Hahn-Banach theorem, is dense in endowed with any topology such that , that is, which respects the duality of and , in particular, for . Since is a Banach space, we can extend it to .
If (ii) is satisfied, then the transpose is weak*-weak continuous, and maps the unit ball of to a relatively compact subset of ([27, 43.3(2),(3)]). We define , since for each , and for each , we have that . We conclude since is in the unit ball of .
Finally, that (iii) implies (i) is trivial.
Observe that, setting in Theorem 19, we obtain a linearization of the space , which also can be obtained as a consequence of the much more general linearization result given by Bierstedt in [14, Bemerkung 3.1] and [15, Corollary 3.94]. Example 15 shows that Theorems 10 and 12 cannot be stated avoiding the condition of relative compactness on the range for general Banach spaces of holomorphic functions.
We finish showing that the weak definition given in this section for is consistent with the natural one, that is,
We use a similar argument to the one used by Bierstedt in [13, page 200] in a more general setting, including our case when is finite dimensional (i.e., putting compact instead of -bounded in the definition of ). If satisfies that for each , then by the previous theorem, there exists defined by which is weak*-weak continuous and such that is relatively compact. This implies that the restriction of to is weak*-norm continuous. Let . There exists a weak* 0-neighbourhood in such that for every . Let be such that . There exists a -bounded subset such that for each . This yields that , and consequently, for every .
The author wants to thank J. Bonet for several references, discussions, and ideas provided, which were very helpful and in particular allowed him to prove Theorem 7, Proposition 8, and Examples 15 and 16. Remark 4 is due to him. The participation of M. J. Beltrán in a lot of discussions during all the work has also been very important. Her ideas are also reflected in the paper. The author is also indebted to L. Frerick and J. Wengenroth for communicating to him Lemma 2. The remarks and corrections of the referee have been also really helpful to the final version. The author thanks him/her for that. This research was partially supported by MEC and FEDER Project MTM2010-15200, GV Project ACOMP/2012/090, and Programa de Apoyo a la Investigacin y Desarrollo de la UPV PAID-06-12.
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