Abstract and Applied Analysis

Volume 2014 (2014), Article ID 612304, 12 pages

http://dx.doi.org/10.1155/2014/612304

Research Article

Homomorphisms between Algebras of Holomorphic Functions

Verónica Dimant,¹ Domingo García,² Manuel Maestre,² and Pablo Sevilla-Peris³

¹Departamento de Matemática, Universidad de San Andrés, Vito Dumas 284, Victoria, B1644BID Buenos Aires, Argentina

²Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, Burjasot, 46100 Valencia, Spain

³Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022 Valencia, Spain

Received 26 December 2013; Accepted 13 March 2014; Published 12 May 2014

Academic Editor: Alfredo Peris

Copyright © 2014 Verónica Dimant et al. 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.

Abstract.

For two complex Banach spaces and , in this paper, we study the generalized spectrum of all nonzero algebra homomorphisms from , the algebra of all bounded type entire functions on , into . We endow with a structure of Riemann domain over whenever is symmetrically regular. The size of the fibers is also studied. Following the philosophy of (Aron et al., 1991), this is a step to study the set of all nonzero algebra homomorphisms from into of bounded holomorphic functions on the open unit ball of and of all nonzero algebra homomorphisms from into .

1. Introduction

The study of homomorphisms between topological algebras is one of the basic issues in this theory. Two are the main topological algebras that we come across when we deal with holomorphic functions on infinite dimensional spaces (see Section 2 for precise definitions): , the holomorphic functions of bounded type (which is a Fréchet algebra), and , the bounded holomorphic functions on the open unit ball (which is a Banach algebra). Here, as a first step in the study of the set of homomorphisms between spaces, we mainly focus on algebras of holomorphic functions of bounded type and homomorphisms between them; (i.e., continuous, linear, and multiplicative mappings). These were already considered in [1]. There the focus was to study the homomorphisms as “individuals,” seeking properties of single ones. We have here a different interest: we treat them as a whole, considering the set

We will call this set the generalized spectrum or simply the spectrum. Our main aim is to study and to define on it a topological and a differential structure.

This problem has the same flavor as considering , the spectrum of the algebra (i.e., the set of nonzero continuous, linear, and multiplicative ). This was studied in [2, 3], where a structure of Riemannian manifold over the bidual was defined on it (see also [4, Section 3.6] for a very neat and nice presentation and [5–7] for similar results). Our approach is very much indebted to that in [2] and we get up to some point analogous results, defining on a Riemann structure over (note that ). We will also be interested in the fibers of elements of .

The outline of the paper is the following. In Section 3, for two complex Banach spaces and , we study the generalized spectrum of all nonzero algebra homomorphisms from to . We endow it with a structure of Riemann domain over whenever is symmetrically regular. In Section 4, we focus on the sets (fibers) of elements in that are projected on the same element of . The size of these fibers is studied and we prove that they are big by showing that they contain big sets. Following the philosophy of [2], all about is a step to study in Section 5 the spectrum of all nonzero algebra homomorphisms from to of bounded holomorphic functions on the open unit ball of . Finally, in Section 6, we deal with the generalized spectrum of all nonzero algebra homomorphisms from to .

2. Definitions and Preliminaries

Unless otherwise stated capital letters such as will denote complex Banach spaces. The dual will be denoted by and the open ball of center and radius by . If and we just write . The space of continuous, linear operators from to will be denoted by ; this is a Banach space with the norm . The adjoint operator of will be denoted by .

We will denote the canonical inclusion of a space into its bidual by .

Given two complex locally convex spaces and , a mapping is a continuous -homogeneous polynomial if there is a continuous -linear mapping such that

Throughout the paper every polynomial and multilinear mapping will be assumed to be continuous.

An -linear mapping is called symmetric if for every permutation of . Each -homogeneous polynomial has a unique symmetric mapping (which we denote by ) satisfying (2). If is a continuous -homogeneous polynomial between Banach spaces, the following expressions define norms for -linear mappings and for -homogeneous polynomials, respectively:

The polarization formula gives [4, Corollary 1.8]

Given an -linear mapping , the notation will mean that is repeated times and is repeated times.

A mapping is holomorphic on the open set if for every in , there exist , each one of them an -homogeneous polynomial, such that the series

converges uniformly in some neighborhood of contained in . This is called the “Taylor series expansion” of at .

If and are Fréchet spaces, then is Gâteaux holomorphic if for every the function , is holomorphic on some neighborhood of . It is known [4, page 152-153] that is holomorphic if and only if it is Gâteaux holomorphic and continuous.

In the case that , and being complex Banach spaces, is holomorphic on if and only if it is Fréchet differentiable on and in that case for every . Also, by the Cauchy inequalities [4, Proposition 3.2] we have, for every ,

A holomorphic function is of bounded type if it sends bounded subsets of to bounded sets of . We denote by the space of holomorphic functions of bounded type from to . If we simply write . Given we write

With this notation for a Banach space is a Fréchet algebra endowed with the topology of uniform convergence on the bounded sets, whose seminorms are (for )

Let be a differential manifold on a complex Banach space and a locally convex space. A mapping is said to be holomorphic (of bounded type) if is holomorphic (of bounded type) for every chart of .

Given , we write for the evaluation mapping at ; that is, for all holomorphic .

Let be a continuous bilinear form. Fix and for let be a net in weak-star convergent to . Since , then there exists . Now, fix and for let be a net in weak-star convergent to . Since , then there exists

Following this idea Aron and Berner showed in [8] that every function admits an extension to the bidual, called the Aron-Berner extension, . By [9, Theorem 3], for every -homogeneous polynomial , we have .

A Banach space is symmetrically regular if for all continuous symmetric bilinear form it follows that

We refer the reader to [4, 10, 11] for the general background on the theory of holomorphic functions on infinite dimensional spaces.

We are going to work with the set of nonzero algebra homomorphisms between spaces of holomorphic functions of bounded type defined in (1). Observe that an idempotent element in satisfies that is a subset of and is a connected set. Hence, either or . So, for any , we should have .

3. The Differential Structure of

Our aim in this section is to endow with a structure of Riemann domain over . On a first step, for each linear operator , we define, inspired by [12, Lemma 1],

It is plain that is an algebra homomorphism and . This defines a natural inclusion:

On the other hand there is also a projection:

These clearly satisfy for every .

Given we consider the mapping defined by . It is a well-known fact that fixed in the mapping given by is a holomorphic function of bounded type. See [4, Proposition 6.30] or more in general the proof of [3, Theorem 2.2].

With this, for each and each , we can define by

for all and . Let us see that is well defined. To do that we need to check that the function

belongs to . We see first that it is holomorphic. The following function of two variables

is holomorphic on each variable. Then Hartogs’ theorem gives that it is holomorphic and hence it is so when restricted to the diagonal . This gives that (15) is holomorphic.

We see now that is of bounded type. Given there exists such that

for all . Hence

since for all and all .

On the other hand it is easy to see that is an algebra homomorphism, which finally gives .

As a second step we show that

Indeed, for each and , we have

Thus,

Finally, given and , we consider the set

Now we assume that is symmetrically regular. We proceed as in the definition of the Riemann domain structure of the spectrum of and we omit the details (see [4, Section 6.3] for a complete and detailed explanation of the procedure). First of all, if , then , with . Hence, for any ,

Therefore, for , we have and is a neighborhood basis at .

Also, for , we have that if , then , for all and if , then for . This gives that the topology generated by is Hausdorff.

Let us note that for each in the subset is the connected component containing .

Summing all this up we have proved the following result.

Proposition 1. If is a symmetrically regular Banach space and is any Banach space, is a Riemann domain over and each connected component of is homeomorphic to .

Our aim now is to show that each function can be extended, in some sense, to a function on of bounded type. We do it with the following sort of Gelfand transform:

and showing that this, when restricted to each connected component, is holomorphic of bounded type. To do that we need the following lemma.

Lemma 2. If and are complex Banach spaces and is an element of , then the mapping defined by for belongs to . Conversely, given in the mapping belongs to .

Proof. Let be in , and let be the Taylor series expansion of at . We have

Note that if we take by the polarization formula (4) and Cauchy inequalities (6), we have

By the properties of convergent double series of nonnegative numbers we obtain that, for each fixed , the series converges absolutely and uniformly on any product of balls in and with finite radii. Hence is a continuous -homogeneous polynomial and actually is an entire function from to that, by above inequalities, is of bounded type.

Conversely, consider in and define by . By definition, for each , belongs to . If we fix now , we have that is a continuous linear form on and , implying that is the composition of holomorphic mappings. Thus is holomorphic for every . By Hartogs’ theorem, . Finally, for fixed we have that ) is a bounded subset of . Hence

and is bounded on the bounded subsets of .

Proposition 3. Let be a symmetrically regular Banach space and let be any Banach space. Given a function consider its extension defined in (24). We have that is a holomorphic function of bounded type. That is, for every .

Proof. The point is to prove that the function

is holomorphic of bounded type. For that, we introduce an auxiliary mapping , defined by

As above, we only need to check that it is separately holomorphic to conclude that is holomorphic. First we fix and and denote . We have and this belongs to . Now we fix and take . This mapping is holomorphic (of bounded type) since it is the composition of the linear mapping defined by with the holomorphic mapping of bounded type:

Finally we fix and denote . Again, this is the composition of a linear mapping defined by with the same holomorphic mapping (30). We conclude that is holomorphic.

Let now . Given there exists such that for every . Hence

and is of bounded type. Since the mapping defined by is obviously holomorphic of bounded type we have that

defined by is holomorphic of bounded type. Now a direct application of Lemma 2 gives that the mapping belongs to .

The above proposition is related to the study of extension of functions of bounded type given in [13].

4. The Size of the Fibers of

We focus now on the sets of elements in that are projected on the same element of . This is called the “fiber” of and is defined by

Our aim in this section is to find out how big these fibers can be. To begin with, each fixed defines a set

On the other hand every defines a composition homomorphism given by . This gives an inclusion

which maps the set into the fiber . Let us check that is injective. Given , , there exists such that and since separates points of we can find with . Thus and this gives .

There is also a projection

The mapping belongs to (and hence is well defined). This follows from the fact that

is holomorphic. Clearly, . Also, note that determines the values that takes when restricted to . This means that when finite type polynomials are dense in , determines . So, the only homomorphisms in are the ’s and we have the following result, which is closely related to [13, Lemmas 4.5 and 4.6] and in [1, Theorem 21].

Proposition 4. Let and be Banach spaces. If finite type polynomials are dense in then for each there exists such that . Also, .

Let us note that the mapping is actually injective and, by Proposition 4, if finite type polynomials are dense, surjective. This means that even in the case when finite type polynomials are dense in (i.e., the space is rather small), the fibers are thick. Let us see now that if this is not the case (i.e., when there is a polynomial in that is not weakly continuous on bounded sets), this mapping is no longer surjective and we can find even more homomorphisms inside each fiber.

Proposition 5. If is symmetrically regular and there exists a polynomial on not weakly continuous on bounded sets at a point , then, for each , there is such that , for all .

Proof. It is enough to prove the result for because we can change fibers through the mapping

Also, if , then , with .

Let be a polynomial that is not weakly continuous on bounded sets at and a bounded net weakly convergent to such that , for all . For an ultrafilter containing the sets , let be given by

Then is a homomorphism in (actually in ), that is, not of composition type. Indeed, since is a constant function on it follows that for every in and so . If for certain , then we have

This says that , for all . Hence,

which is a contradiction.

Something more can be said when there is a polynomial that is not weakly continuous on bounded sets. In this case, we can insert in each fiber a big set of homomorphisms that are not of composition type. We do it in detail in the fiber of 0.

First, note that if there is a polynomial not weakly continuous on bounded sets, then there is a homogeneous polynomial which is not weakly continuous on bounded sets at 0. So, the Aron-Berner extension of is not -continuous at any point [14, Corollary 2, Proposition 3] and [15, Proposition 1] (see also [16, Proof of Proposition 2.6]). Thus, we fix, for each a bounded net in which -converges to , but , for all . For each we fix also an ultrafilter containing the sets . Consider the set

and define the following mapping

where

This mapping is well defined because ; then . The mapping is also injective. Indeed, if , then , for all and . Then,

Therefore, , for all and, evaluating at 0, we obtain that and, hence, .

Note, also, that the homomorphisms are not of composition type. Indeed, if , for certain , , and , then, for all ,

If this were the case we would have , for all . Now,

Evaluating at 0 leads to a contradiction.

Let us finish this analysis by studying : the composition of the inclusion defined in (35) with the inclusion of into . Then the mapping turns out to be holomorphic (endowing with an appropriate topology). We prepare the proof of this fact with a lemma that is a variant of a classical Dunford result; see also [17, Theorem 3].

Lemma 6. let be a Gâteaux holomorphic mapping from a complex Fréchet space to a Banach space such that is holomorphic for every , where is a norming subset of the closed unit ball of (i.e., for every ). Then is holomorphic on .

Proof. Let be a finite dimensional compact subset of . By hypothesis, is a compact subset of and hence it is bounded. Thus, there exists such that

Hence, the family of scalar valued holomorphic functions is bounded on the finite dimensional compact subsets of . But as is a Fréchet space, then it is also Baire, and by [11] this family is locally bounded: given there exists an open neighborhood of such that

We have obtained that the Gâteaux holomorphic function is locally bounded. Then it is holomorphic by [4, Proposition 3.7].

The following proposition gives that is holomorphic. The proof may seem at some points similar to that of Proposition 3, but the fact that in the target space we have now a nonmetrizable locally convex topology makes the whole situation much more delicate. We are going to consider now in the topology defined by the following fundamental system of seminorms:

for , where and is a bounded subset of .

Proposition 7. The mapping

is injective and holomorphic if we consider in the -topology.

Proof. Clearly is well defined. Our first step is to prove that is Gâteaux holomorphic. Let , , , and . Here we have . If is the Taylor series expansion of at on , then

for every and the convergence is absolute and uniform on the bounded subsets of [8, 9]. Thus

where our last step is simply formal. We are going to concentrate our effort now to show that this formal last equality holds in our setting. If we denote by the closure of , then is a bounded subset of for . Thus there exists such that , for every in and . We fix and we take such that . We have

Hence, by the properties of summability of double series of nonnegative numbers, the double series below is convergent in :

and its sum is again . On the other hand, by using the polarization formula (4), [9, Theorem 3], and Cauchy’s inequalities (6) we get

By applying now (55) we obtain

Since the function is the composition of a continuous multilinear mapping and two holomorphic mappings of bounded type, it is holomorphic of bounded type on . By using (57), we obtain that the series

-converges in ; hence belongs to for every in . Actually, if we consider , this is a linear operator. By (57), it is also continuous, since given there exists such that

for every . Then

We have, for with , again by (57),

As a consequence the series defined on with values in , -converges uniformly on the compacts of . Hence it is an entire function. Since we have proved that all series involved converge absolutely, we can apply the reordering of absolutely convergent double series to conclude that the last formal equality of (53) actually holds and then

is an entire function on for every . This gives that is Gâteaux holomorphic.

We fix now , a continuous seminorm of the fundamental system defined in (50) and denote by the completion of the normed space . Given and we define the continuous linear functional

by . Clearly the quotient mapping , defined by , belongs to . On the other hand the set is a norming subset of since

We can consider that remains Gâteaux holomorphic. Thus is Gâteaux holomorphic for every and every . If we show that it is continuous, then it will be holomorphic and, by Lemma 6, we will get that is holomorphic for every seminorm . Since is a complete space, we can conclude that is holomorphic.

Let . Now for fixed and , we consider and we choose such that

As is uniformly continuous on bounded subsets of , for a given , there exists such that if with , then

Let with . Clearly . Thus

This gives that is continuous for every and and completes the proof.

We recall that the topology on is the topology of the pointwise convergence on the points of .

Corollary 8. The mapping

is injective and holomorphic when is endowed with the topology induced by .

5. Some Properties of

When Aron et al. undertook in [2] the study of , the spectra of the Fréchet algebra of holomorphic functions of bounded type they explicitly stated that it was a step to study the spectra of the Banach algebra of bounded holomorphic functions on the open unit ball of (endowed with the supremum norm). Following this philosophy we now study the spectrum consisting of nonzero continuous homomorphisms that we will denote by .

As above we can define

Proposition 9. If is a symmetrically regular Banach space and is any Banach space, then is a Riemann domain over and each connected component of is homeomorphic to .

Proof. The proof follows almost word by word that of Proposition 1. The basis of neighborhoods of a point in is given by , for , where

for all and . The fact that is in follows from a similar argument to that in (18) taking .

Now, as in (24), we can define a Gelfand transform of by

and we can see that this is a holomorphic extension of to .

Proposition 10. Let be a symmetrically regular Banach space and let be any Banach space. Given a function consider its extension defined in (71). Then the restriction of to each connected component of is a holomorphic function of bounded type.

Proof. Like in Proposition 3 it is enough to prove that for each fixed in , the mapping defined as for in is holomorphic of bounded type.

First we check that it is uniformly continuous on any bounded subset of . By definition, there exists such that

for every in . Let . Since is uniformly continuous on , given there exists such that if are in with , then . Consider now in with for and . We have

The last inequality holds because for and .

Now we prove that is Gâteaux holomorphic. Let us denote by the Taylor series expansion of at a point in . Consider in . Since is symmetrically regular we have that

for every in and in . Now we fix and take with and . We have, by using the polarization constants and Cauchy inequalities, that

where . As a consequence

Hence we have that the function is well defined and actually belongs to for every nonnegative integer and every in . Moreover, the equality

holds for every in , in and in . Actually, (76) implies also that the series converges in for each in . Hence,

with in for every and and the series converges in for each in . Now, if we take , we have

To finish we just have to observe that the function belongs to for every . This is an immediate consequence of the fact that , where is the function

which clearly is separately holomorphic. Thus, by Hartogs’ theorem, is holomorphic.

Proceeding now as in Section 4 we are going to see how we can insert big sets into the fibers of . Indeed, given we consider given by . This is well defined, since is bounded in and . This again gives an inclusion

that for each maps the set

into the fiber of . Also in this case we have a projection

Again like in Section 4 the map is holomorphic. The fact that it is also bounded follows immediately from the fact that is a bounded subset in and then is bounded in .

Like in the case of we have that if finite type polynomials are dense in then for each there exists such that .

On the other hand for every nontrivial Banach space , there exists an element in that does not belong to . Indeed, in [18] it is shown that there exists a continuous homomorphism such that , where , and vanishes on all continuous homogeneous polynomial on of odd degree. We define as for . Clearly is in . Observe that if for a certain , then

for every and every , and we have obtained that . Hence for every and every . In particular, , a contradiction.

6. Homomorphisms between Algebras of Bounded Functions

In this last section we introduce, for two given and complex Banach spaces,

In this case, no Riemann manifold structure is known, but on the other hand we are going to show that some results very close to the Gelfand transform of the elements of a uniform algebra can be obtained.

The space is a dual space. That was proved by Mujica in [19], where he found a topological predual that is a subspace of such that if and only if the restriction of to is continuous when endowed with the compact open topology . Let us observe that is a subset of . We denote by the weak-star topology .

Theorem 11. is a -compact subset of  .

Proof. An application of the Banach Steinhaus theorem yields that and coincide as sets.

By Alaoglu-Bourbaki theorem any bounded subset of is weak-star relatively compact. Thus, by [20, 39.4] the space has the property that every equicontinuous subset of it is relatively -compact. As the spectrum is equicontinuous, we obtain that it is relatively -compact. Now we check that the spectrum is -closed. Take in the closure of and let be a net in the spectrum convergent to . Then, given in we have that , , and -converge to , , and , respectively. But the topology in coincides with the topology of uniform convergence on the compact subsets of . On the other hand in [19, 2.1 Theorem] it is proved that the mapping

defined by is holomorphic. Hence if is a compact subset of , then is a compact subset of and we obtain that converges uniformly to on . Also converges uniformly to on . As a consequence the net converges to on the compact open topology of . This implies, by the definition of , that converges to for every in . In other words the net -converges to and we have

for every in . Thus belongs to and the conclusion follows.

In our setting, taking as a -compact set, we are going to extend the concept of Gelfand transform of an element of a Banach algebra in the following way. Given , define

by .

Proposition 12. The mapping

is an isometry of algebras.

Proof. Clearly it is an algebra homomorphism and

for every in . For a fixed , given there exists such that . Let and consider any with that attains its norm at a certain with . Let be defined by . We have that belongs to and

Hence the inequality also holds.

The scalar spectrum of , denoted by , can be considered as a subset of by associating to each in the homomorphism that belongs to . In that way the new Gelfand transform of each in can be considered and extension of the classical by the equality

It is natural to consider again and under the light of Proposition 12. We cannot give to any of these two spectra a topology that makes them compact sets. But in the case of it can be endowed with a topology in such a way that this set is a countable union of compact sets. Indeed, as above, an application of the Banach Steinhaus theorem, now for Fréchet spaces, yields that and coincide as sets and, also as above, any equicontinuous subset of is relatively -compact. Given , we denote

Then, is an equicontinuous set, and a similar argument of one given in the proof of Proposition 12 implies that is a compact set. Finally .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was initiated in June 2012, when the first named author visited the Universidad de Valencia and the Universitat Politècnica de València. She wishes to thank all the people in and outside both universities who made that visit such a delightful time. The first named author was partially supported by research projects CONICET PIP 0624 and ANPCyT PICT 1456. The second, third, and fourth named authors were supported by research project MICINN MTM2011-22417. The second and third named authors also by research project Prometeo II/2013/013.

References

D. Carando, D. García, and M. Maestre, “Homomorphisms and composition operators on algebras of analytic functions of bounded type,” Advances in Mathematics, vol. 197, no. 2, pp. 607–629, 2005.

R. M. Aron, B. J. Cole, and T. W. Gamelin, “Spectra of algebras of analytic functions on a Banach space,” Journal für die Reine und Angewandte Mathematik, vol. 415, pp. 51–93, 1991.

R. M. Aron, P. Galindo, D. García, and M. Maestre, “Regularity and algebras of analytic functions in infinite dimensions,” Transactions of the American Mathematical Society, vol. 348, no. 2, pp. 543–559, 1996.

S. Dineen, Complex Analysis on Infinite-Dimensional Spaces, Springer Monographs in Mathematics, Springer, London, UK, 1999.

M. J. Beltrán, “Spectra of weighted (LB)-algebras of entire functions on Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 387, no. 2, pp. 604–617, 2012.

D. Carando, D. García, M. Maestre, and P. Sevilla-Peris, “A Riemann manifold structure of the spectra of weighted algebras of holomorphic functions,” Topology, vol. 48, no. 2–4, pp. 54–65, 2009.

D. Carando and P. Sevilla-Peris, “Spectra of weighted algebras of holomorphic functions,” Mathematische Zeitschrift, vol. 263, no. 4, pp. 887–902, 2009.

R. M. Aron and P. D. Berner, “A Hahn-Banach extension theorem for analytic mappings,” Bulletin de la Société Mathématique de France, vol. 106, no. 1, pp. 3–24, 1978.

A. M. Davie and T. W. Gamelin, “A theorem on polynomial-star approximation,” Proceedings of the American Mathematical Society, vol. 106, no. 2, pp. 351–356, 1989.

J. Mujica, Complex Analysis in Banach Spaces, vol. 120 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, Holomorphic functions and domains of holomorphy in finite and infinite dimensions, Notas de Matemática [Mathematical Notes], 107, 1986.

J. A. Barroso, M. C. Matos, and L. Nachbin, “On holomorphy versus linearity in classifying locally convex spaces,” in Infinite Dimensional Holomorphy and Applications, vol. 12 of North-Holland Mathematics Studies, pp. 31–74, North-Holland, Amsterdam, The Netherlands, 1977.

S. Lassalle and I. Zalduendo, “To what extent does the dual Banach space determine the polynomials over ?” Arkiv för Matematik, vol. 38, no. 2, pp. 343–354, 2000.

D. Carando and S. Muro, “Envelopes of holomorphy and extension of functions of bounded type,” Advances in Mathematics, vol. 229, no. 3, pp. 2098–2121, 2012.

C. Boyd and R. A. Ryan, “Bounded weak continuity of homogeneous polynomials at the origin,” Archiv der Mathematik, vol. 71, no. 3, pp. 211–218, 1998.

R. Aron and V. Dimant, “Sets of weak sequential continuity for polynomials,” Indagationes Mathematicae, vol. 13, no. 3, pp. 287–299, 2002.

R. M. Aron, D. Carando, S. Lassalle, and M. Maestre, “Cluster values of holomorphic functions of bounded type,” preprint

K.-G. Grosse-Erdmann, “A weak criterion for vector-valued holomorphy,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, no. 2, pp. 399–411, 2004.

D. Deghoul, “Construction de caractères exceptionnels sur une algèbre de Fréchet,” Comptes Rendus de l'Académie des Sciences., vol. 312, no. 8, pp. 579–580, 1991.

J. Mujica, “Linearization of bounded holomorphic mappings on Banach spaces,” Transactions of the American Mathematical Society, vol. 324, no. 2, pp. 867–887, 1991.

G. Köthe, Topological Vector Spaces. II, vol. 237 of Grundlehren der Mathematischen Wissenschaften, Springer, New York, NY, USA, 1979.