Pablo Galindo

University of Valencia | UV

A wide new class of subsets of a Banach space X named coarse p-limited sets (1≤p

A wide new class of subsets of a Banach space $X$ named coarse $p$-limited sets ($ 1\leq p < \infty$) is introduced by considering weak* $p$-summable sequences in $X'$ instead of weak* null sequences. We study its basic properties and compare it with the class of compact and weakly compact sets. Results concerning the relationship of coarse $p$-lim...

In the setting of Banach lattices the weak (resp. positive) Grothendieck spaces have been defined. We localize such notions by defining new classes of sets that we study and compare with some quite related different classes. This allows us to introduce and compare the corresponding linear operators.

We prove that, as in the finite dimensional case, the space of Bloch functions on the unit ball of a Hilbert space contains, under very mild conditions, any semi-Banach space of analytic functions invariant under automorphisms. The multipliers for such Bloch space are characterized and some of their spectral properties are described.

We prove that under some mild conditions on the symbol φ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi ,$$\end{document} the spectrum of the corresponding com...

We consider the subalgebra \(H_{bs}(L_\infty )\) of analytic functions of bounded type on \(L_\infty [0,1]\) which are symmetric, i.e. invariant, with respect to measurable bijections of [0, 1] that preserve the measure. Our main result is that \(H_{bs}(L_\infty )\) is isomorphic to the algebra of all analytic functions on the strong dual of the sp...

We discuss the continuity of the composition on several spaces of holomorphic mappings on open subsets of a complex Banach space. On the Fréchet space of entire mappings that are bounded on bounded sets, the composition turns out to be even holomorphic. In such a space, we consider linear subspaces closed under left and right composition. We discus...

Let E be a complex Banach space with open unit ball \(B_E.\) For analytic self-maps \(\varphi \) of \(B_E\) with \(\varphi (0) =0,\) we investigate the spectra of weighted composition operators \(uC_\varphi \) acting on a large class of spaces of analytic functions. This class contains, for example, weighted Banach spaces of \(H^\infty \)-type on \...

We present an infinite-dimensional version of Cartan's theorem concerning the existence of a holomorphic inverse of a given holomorphic self-map of a bounded convex open subset of a dual Banach space. No separability is assumed, contrary to previous analogous results. The main assumption is that the derivative operator is power bounded, and which w...

We show that an interpolating sequence for the weighted Banach space of analytic functions on the unit ball of a Hilbert space is hyperbolically separated. In the case of the so-called standard weights, a sufficient condition for a sequence to be linear interpolating is given in terms of Carleson type measures. Other conditions to be linearly inter...

The spectra and essential spectra of some weighted composition operators arising from non-automorphic parabolic self-maps of the unit disc acting on a class of Banach spaces- not necessarily reflexive- of analytic functions is shown to be a spiral-like set, as it happens in the cases treated earlier.

We consider on the space ℓ∞ polynomials that are invariant regarding permutations of the sequence variable or regarding finite permutations. Accordingly, they are trivial or factor through c0. The analogous study, with analogous results, is carried out on L∞[0,+∞), replacing the permutations of N by measurable bijections of [0,+∞) that preserve the...

In this paper we estimate the norm and the essential norm of weighted composition operators from a large class of non-necessarily reflexive Banach spaces of analytic functions on the open unit disk into weighted type Banach spaces of analytic functions and Bloch type spaces. We also show the equivalence of compactness and weak compactness of weight...

We consider the algebra of holomorphic functions on L∞ that are symmetric , i.e. that are invariant under composition of the variable with any measure-preserving bijection of [0, 1]. Its spectrum is identified with the collection of scalar sequences such that is bounded and turns to be separable. All this follows from our main result that the subal...

In this paper we estimate the norm and the essential norm of weighted composition operators from a large class of- non-necessarily reflexive- Banach spaces of analytic functions on the open unit disk into weighted type Banach spaces of analytic functions and Bloch type spaces. We also show the equivalence of compactness and weak compactness of weig...

An extension of Lempert’s result about non approximability by entire functions of analytic functions on some open subsets of ℓ∞ is obtained for Banach spaces having a bounding non relatively compact set.We also prove that subsets A that are bounding for analytic functions defined in any of its neighborhoodswhose boundary lies at positive distance f...

We study the holomorphic functions on a complex Banach space $E$ that are invariant under the action of a given group of operators on $E$. A great variety of situations occur depending, of course, on the group and the space. Nevertheless, in the examples we deal with, they can be described in terms of a few natural ones and functions of a finite nu...

We discuss when the Königs eigenfunction associated with a non-automorphic selfmap of the complex unit disc that fixes the origin belongs to Banach spaces of holomorphic functions of Bloch and H∞ type. In the latter case, our characterization answers a question of P. Bourdon. Some spectral properties of composition operators on H∞ for unbounded Kön...

An extension of Lempert’s result about non approximability by entire functions of analytic functions on some open subsets of ℓ∞ is obtained for Banach spaces having a bounding non relatively compact set. We also prove that subsets A that are bounding for analytic functions defined in any of its neighborhoodswhose boundary lies at positive distance...

Every analytic self-map of the unit ball of a Hilbert space induces a bounded composition operator on the space of Bloch functions. Necessary and sufficient conditions for compactness of such composition operators are provided, as well as some examples that clarify the connections among such conditions.

The Bloch space has been studied on the open unit disk of CC and some homogeneous domains of CnCn. We define Bloch functions on the open unit ball of a Hilbert space E and prove that the corresponding space B(BE)B(BE) is invariant under composition with the automorphisms of the ball, leading to a norm that — modulo the constant functions — is autom...

We consider the uniform algebra A(BX) of continuous and bounded functions that are analytic on the interior of the closed unit ball BX of a complex Banach function module X. We focus on norming subsets of BX, i.e., boundaries, for such algebra. In particular, if X is a dual complex Banach space whose centralizer is infinite-dimensional, then the in...

In this paper we introduce and study the notion of homogeneous Tauberian polynomial, aiming at extending the concept of Tauberian operator. Such notion is characterized in terms of the polynomial topology for which we prove a Banach-Alaoglu type theorem. A number of examples show that the behavior of Tauberian polynomials differs from that of Taube...

We provide two function-theoretic estimates for the essential norm of a composition operator CφCφ acting on the space BMOA; one in terms of the n-th power φnφn of the symbol φ and one which involves the Nevanlinna counting function. We also show that if the symbol φ is univalent, then the essential norm of CφCφ is comparable to its essential norm o...

We present an infinite-dimensional version of Cartan-Carathéodory-Kaup-Wu theorem about the analyticity of the inverse of a given analytic mapping. It is valid for a class of domains in separable Banach dual spaces that includes all bounded convex domains.

On natural uniform Banach algebras of bounded analytic functions defined either in the infinite-dimensional Euclidean ball or the infinite-dimensional polydisc, we discuss Gleason’s problem whether the ideal of functions that vanish at some given point is generated by the canonical projections. Sufficient conditions for an endomorphism to be a comp...

We show that the spectrum of the algebra of bounded symmetric analytic functions on ℓp,1≤p<+∞ with the symmetric convolution operation is a commutative semigroup with the cancellation law for which we discuss the existence of inverses. For p=1p=1, a representation of the spectrum in terms of entire functions of exponential type is obtained which al...

We give a very short proof of a characterization of Fredholm composition operators acting on a large class of Banach spaces of analytic functions defined on the open unit ball of the Euclidian n-complex space.

In the spectrum of the algebra of symmetric analytic functions of bounded type on ℓp, 1 ≤ p < +∞, and along the same lines as the general non-symmetric case, we define and study a convolution operation and give a formula for the ‘radius’ function. It is also proved that the algebra of analytic functions of bounded type on ℓ1 is isometrically isomor...

The essential norm of any operator from a general Banach space of holomorphic functions on the unit ball in ℂ𝑛 into the little weighted-type space is calculated. Some applications of the formula are given.

The theme of this paper is the study of the separability of subspaces of holomorphic functions respect to the convergence over a given set and its connection with the metrizability of the polynomial topology. A notion closely related to this matter is that of Asplund set. Our discussion includes an affirmative answer to a question of Globevnik abou...

We obtain a formula for the essential norm of any operator be-tween weighted Bergman spaces of infinite order. Then we apply it to obtain or estimate essential norms of operators acting on Bloch type spaces and to differences of composition operators or Toeplitz operators on some weighted Bergman spaces. KEYWORDS: Weighted Bergman spaces of infinit...

We prove that Fredholm composition operators acting on the uniform algebra H∞(BE) of bounded analytic functions on the open unit ball of a complex Banach space E with the approximation property are invertible and arise from analytic automorphisms of the ball.

Homogeneous polynomials of degree 2 on the complex Banach space c0(ℓn2) are shown to have unique norm-preserving extension to the bidual space. This is done by using M-projections and extends the analogous result for c0 proved by P.-K. Lin. © 2009. Published by Oxford University Press. All rights reserved.

We continue the study of spectra of non power-compact composition operators on H¥(BE)H^\infty(B_{E}), E a complex Banach space. We obtain a Julia-type estimate of the growth of the symbol near the sphere for E =C0(X), thus for c0 or \mathbbCn{\mathbb{C}}^n. In particular, the description of the spectra of non power-compact composition operators o...

We consider the problem of whether a given interpolating sequence for a uniform algebra yields linear interpolation. A positive answer is obtained when we deal with dual uniform algebras. Further we prove that if the Carleson generalized condition is sufficient for a sequence to be interpolating on the algebra of bounded analytic functions on the u...

Let E be a Banach space, with unit ball B E . We study the spectrum and the essential spectrum of a composition operator on H ∞ (BE) determined by an analytic symbol with a fixed point in BE. We relate the spectrum of the composition operator to that of the derivative of the symbol at the fixed point. We extend a theorem of Zheng to the context of...

For each ideal of multilinear mappings $\mathcal{M}$ we explicitly construct a corresponding ideal ${}^{a}\mathcal{M}$ such that multilinear forms in ${}^{a}\mathcal{M}$ are exactly those which can be approximated, in the uniform norm, by multilinear forms in $\mathcal{M}$. This construction is then applied to finite type, compact, weakly compact a...

We determine the spectra of composition operators acting on weighted Banach spaces Hv∞ of analytic functions on the unit disc defined for a radial weight v, when the symbol of the operator has a fixed point in the open unit disc. We also investigate in this case the growth rate of the Koenigs eigenfunction and its relation with the essential spectr...

We discuss a strong version of the Dunford-Pettis property, earlier named (DP*) property, which is shared by both ℓ1 and ℓ∞. It is equivalent to the Dunford-Pettis property plus the fact that every quotient map onto CQ is completely continuous. Other weak sequential continuity results on polynomials and analytic mappings related to the (DP*) proper...

We prove that any sequence in the open ball of a complex Banach space E, even in that of E , whose norms are an interpolating sequence for H 1 , is interpolating for the space of all bounded analytic functions on BE. The construction made yields that the interpolating functions depend linearly on the interpolated values.

Let A be a uniform algebra, and let φ be a self-map of the spectrum $M_{A}$ of A that induces a composition operator $C_{\phi}$ on A. The object of this paper is to relate the notion of "hyperbolic boundedness" introduced by the authors in 2004 to the essential spectrum of $C_{\phi}$ . It is shown that the essential spectral radius of $C_{\phi}$ is...

We discuss necessary conditions for a Banach space to satisfy the property that its bounded sets are bounding in its bidual space. Apart from the classic case of c 0 , we prove that, among others, the direct sum c 0 (l n 2) is another example of spaces having such property. A subset B of a complex Banach space X is said to be bounding if every enti...

We study the class of Banach algebra-valued n-homogeneous polynomials generated by the nth powers of linear operators. We compare it with the finite type polynomials. We introduce a topology $w_{EF}$ on E, similar to the weak topology, to clarify the features of these polynomials.

We study the class of Banach algebra-valued n n -homogeneous polynomials generated by the n t h n^{th} powers of linear operators. We compare it with the finite type polynomials. We introduce a topology w E F w_{EF} on E , E, similar to the weak topology, to clarify the features of these polynomials.

We define the class of integral holomorphic functions over Banach spaces; these are functions admitting an integral representation akin to the Cauchy integral formula, and are related to integral polynomials. After studying various properties of these functions, Banach and Fréchet spaces of integral holomorphic functions are defined, and several as...

Let A be a uniform algebra, and let φ be a self-map of the spectrum MA of A that induces a composition operator Cφ on A. It is shown that the image of MA under some iterate φn of φ is hyperbolically bounded if and only if φ has a finite number of attracting cycles to which the iterates of φ converge. On the other hand, the image of the spectrum of...

Weakly compact homomorphisms between (URM) algebras with connected maximal ideal space are shown to factor through H∞(D) by means of composition operators and to be strongly nuclear. The spectrum of such homomorphisms is also described. Strongly nuclear composition operators between algebras of bounded analytic functions are characterized. The path...

LetE be a complex Banach space with open unit ballB e. The structure of the space of composition operators on the Banach algebra H∞, of bounded analytic functions onB e with the uniform topology, is studied. We prove that the composition operators arising from mappings whose range lies strictly insideB e form a path connected component. WhenE is a...

We study the algebra of uniformly continuous holomorphic symmetric functions on the ball of ℓp, investigating in particular the spectrum of such algebras. To do so, we examine the algebra of symmetric polynomials on ℓp− spaces as well as finitely generated symmetric algebras of holomorphic functions. Such symmetric polynomials determine the points...

We study homomorphisms between Fréchet algebras of holomorphic functions of bounded type. In this setting we prove that any pointwise bounded homomorphism into the space of entire functions of bounded type is rank one. We characterize up to the approximation property of the underlying Banach space, the weakly compact composition operators on Hb(V),...

The weak compactness of the composition operator C Φ ( f ) = f ○ Φ acting on the uniform algebra of analytic uniformly continuous functions on the unit ball of a Banach space with the approximation property is characterized in terms of Φ. The relationship between weak compactness and compactness of these composition operators and general homomorphi...

. We study the algebra of uniformly continuous holomorphic symmetric functions on the ball of # p , investigating in particular the spectrum of such algebras. To do so, we examine the algebra of symmetric polynomials on # p - spaces as well as finitely generated symmetric algebras of holomorphic functions. Such symmetric polynomials determine the p...

This paper contains characterizations of the bidual space of some closed subspaces of Hb(U), the space of holomorphic functions of bounded type defined on an open subset U of a Banach space X, where U is either a bounded balanced open set or the whole space X.

We prove a characterization (up to the approximation property) of weakly compact,composition operators C : H,(BE )i n terms of their inducing analytic maps,: BE ! BF . Let E denote a complex Banach space with open unit ball BE and let,: BE !

 If points in nontrivial Gleason parts of a uniform Banach algebra have unique representing measures, then the weak and the norm topology coincide on the spectrum. We derive from this several consequences about weakly compact homomorphisms and discuss the case of other uniform Banach algebras arising in complex infinite dimensional analysis.

We show that weak holomorphic convergence and norm convergence coincide for sequences in a Banach space E if and only if every bounding subset of E is relatively compact.

We prove that Gleason parts in uniform distinguished Fréchet algebras are open and closed in the weak topology.

The notion of Arens regularity of a bilinear form on a Banach space E is extended to continuous m-linear forms, in such a way that the natural associated linear mappings, E→L (m−1E) and (m – l)-linear mappings E × … × E → E', are all weakly compact. Among other applications, polynomials whose first derivative is weakly compact are characterized.

A Banach space E is known to be Arens regular if every continuous linear mapping from E to E′ is weakly compact. Let U be an open subset of E, and let Hb(U) denote the algebra of analytic functions on U which are bounded on bounded subsets of U lying at a positive distance from the boundary of U. We endow Hb(U) with the usual Fréchet topology.Mb(U)...

We prove that scalar-valued polynomials are weakly continuous on limited sets and that, as in the case of linear mappings, every c0-valued polynomial maps limited sets into relatively compact ones. We also show that a scalar-valued polynomial whose derivative is limited is weakly sequentially continuous.

In [13] Mazet proved the following result. If U is an open subset of a locally convex space E then there exists a complete locally convex space ( U ) and a holomorphic mapping δ U : U → ( U ) such that for any complete locally convex space F and any f ɛ ℋ ( U;F ), the space of holomorphic mappings from U to F , there exists a unique linear mapping...

By following an idea of O. Nicodemi [Lect. Notes Math. 843, 534-546 (1981; Zbl 0482.46012)], we study certain sequences of extension operators for multilinear mappings on Banach spaces starting from any given extension operator for linear mappings. In this way, we obtain several new properties of the extension operators previously studied by Aron,...

We show that holomorphic mappings of bounded type defined on Fréchet spaces extend to the bidual. The relationship between holomorphic mappings of bounded type and of uniformly bounded type is discussed and some algebraic and topological properties of the space of all entire mappings of (uniformly) bounded type are proved, for example a holomorphic...

We study the holomorphic functions of bounded type defined on (DF)-spaces. We prove that they are of uniformly bounded type. The space of all these functions is a Fréchet space with its natural topology. Some consequences and related results are obtained.

For a Banach space E, we prove that the Fréchet space b(E) is the strong dual of an (LB)-space, b(E), which leads to a linearization of the holomorphic mappings of bounded type. It is also shown that the holomorphic functions defined on (DFC)-spaces are of uniformly bounded type.

We study topological properties of the space (ℋ(E),τ δ ) for a fully nuclear space E with basis. This leads us to a negative solution of a problem of Mujica and Nachbin concerning the relationship between properties of holomorphic mappings defined on a locally convex space E and linear properties of a predual of ℋ(E). The solution is provided by th...

Several results and examples about locally bounded sets of holomorphic mappings defined on certain classes of locally convex spaces (Baire spaces, $(DF)$-spaces, $C(X)$-spaces) are presented. Their relation with the classification of locally convex spaces according to holomorphic analogues of barrelled and bornological properties of the linear theo...

The holomorphically ultrabornological spaces are introduced. Their relation with other holomorphically significant classes of locally convex spaces is established and separating examples are given. Some apparently new properties of holomorphically barrelled spaces are included and holomorphically ultrabornological spaces are utilized in a problem p...

This paper is devoted to obtaining sequence space representations of spaces of vector-valued C k -functions defined on an open subset, Ω, of ℝ ⁿ , whose k th derivatives satisfy a Lipschitz condition on compact subsets of Ω.

In this paper we give a representation of the spaces $C_o^k(\Omega,E)$ and prove that the space $C^k(I,E)$ has certains properties of complementation. Other related spaces are studied.