A Banach spaceE is known to be Arens regular if every continuous linear mapping fromE toE0 is wea... more A Banach spaceE is known to be Arens regular if every continuous linear mapping fromE toE0 is weakly compact. LetU be an open subset ofE, and letHb(U) denote the algebra of analytic functions onU 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 Fr echet
In [13] Mazet proved the following result. If U is an open subset of a locally convex space E the... more 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 5 U :U^(S(U) such that for any complete locally convex space F and any / e%C{U\F), the space of holomorphic mappings from U to F, there exists a unique linear mapping 7}: ^(U)-* F such that the following diagram commutes;
Journal für die reine und angewandte Mathematik (Crelles Journal), 2000
Page 1. J. reine angew. Math. 557 (2003), 173197 Journal für die reine und angewandte Mathematik... more Page 1. J. reine angew. Math. 557 (2003), 173197 Journal für die reine und angewandte Mathematik ( Walter de Gruyter Berlin Á New York 2003 Bohr's power series theorem and local Banach space theory To the memory of our friend Klaus Floret ...
We obtain general lower and upper estimates for the first and the second Bohr radii of bounded co... more We obtain general lower and upper estimates for the first and the second Bohr radii of bounded complete Reinhardt domains in C n : r 2004 Elsevier Inc. All rights reserved.
THE COINCIDENCE OF T0 AND TC,, FOR SPACES OF HOLOMORPHIC FUNCTIONS ON SOME FRECHET-MONTEL SPACES ... more THE COINCIDENCE OF T0 AND TC,, FOR SPACES OF HOLOMORPHIC FUNCTIONS ON SOME FRECHET-MONTEL SPACES ... By PABLO GALINDO, DOMINGO GARCIA and MANUEL MAESTRE ... Departamento de Andlisis Matemaitico, Universitat de Valencia ... [Received 26 March ...
Journal of Mathematical Analysis and Applications, 2014
ABSTRACT We characterize the Banach spaces Y for which certain subspaces of operators from L1(μ)L... more ABSTRACT We characterize the Banach spaces Y for which certain subspaces of operators from L1(μ)L1(μ) into Y have the Bishop–Phelps–Bollobás property in terms of a geometric property of Y, namely AHSP. This characterization applies to the spaces of compact and weakly compact operators. New examples of Banach spaces Y with AHSP are provided. We also obtain that certain ideals of Asplund operators satisfy the Bishop–Phelps–Bollobás property.
Journal of Mathematical Analysis and Applications, 2014
ABSTRACT In this paper we study some geometrical properties of certain classes of uniform algebra... more ABSTRACT In this paper we study some geometrical properties of certain classes of uniform algebras, in particular the ball algebra image of all uniformly continuous functions on the closed unit ball and holomorphic on the open unit ball of a complex Banach space image. We prove that image has image-numerical index 1 for every image, the lushness and also the AHSP. Moreover, the disk algebra image, and more in general any uniform algebra whose Choquet boundary has no isolated points, is proved to have the polynomial Daugavet property. Most of those properties are extended to the vector valued version image of a uniform algebra image.
Given an entire mapping f ∈ H b (X, X) of bounded type from a Banach space X into X, we denote by... more Given an entire mapping f ∈ H b (X, X) of bounded type from a Banach space X into X, we denote by f the Aron-Berner extension of f to the bidual X * * of X. We show that g • f = g • f if X is symmetrically regular. We also give a counterexample on 1 such that the equality does not hold. We prove that the closure of the numerical range of f is the same as that off .
If X is an Asplund space, then every uniformly continuous function on B X * which is holomorphic ... more If X is an Asplund space, then every uniformly continuous function on B X * which is holomorphic on the open unit ball, can be perturbed by a w * continuous and homogeneous polynomial on X * to obtain a norm attaining function on the dual unit ball. This is a consequence of a version of Bourgain-Stegall's variational principle. We also show that the set of N -homogeneous polynomials between two Banach spaces X and Y whose transposes attain their norms is dense in the corresponding space of N -homogeneous polynomials. In the case when Y is the space of Radon measures on a compact K, this result can be strengthened.
Journal of Mathematical Analysis and Applications, 1987
The holomorphically ultrabornological spaces are introduced. Their relation with other holomorphi... more 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 posed by Nachbin.
Proceedings of the American Mathematical Society, 2007
We prove a multilinear version of Phelps' Lemma: if the zero sets of multilinear forms of norm on... more We prove a multilinear version of Phelps' Lemma: if the zero sets of multilinear forms of norm one are 'close', then so are the multilinear forms.
Each Dirichlet series $D = \sum_{n=1}^{\infty} a_n \frac{1}{n^s}$, with variable $s \in \mathbb{C... more Each Dirichlet series $D = \sum_{n=1}^{\infty} a_n \frac{1}{n^s}$, with variable $s \in \mathbb{C}$ and coefficients $a_n \in \mathbb{C}$, has a so called Bohr strip, the largest strip in $\mathbb{C}$ on which $D$ converges absolutely but not uniformly. The classical Bohr-Bohnenblust-Hille theorem states that the width of the largest possible Bohr strip equals $1/2$. Recently, this deep work of Bohr, Bohnenblust and Hille from the beginning of the last century was revisited by various authors. New methods from different fields of modern analysis (e.g. probability theory, number theory, functional and Fourier analysis) allow to improve the Bohr-Bohnenblust-Hille cycle of ideas, and to extend it to new settings, in particular to Dirichlet series which coefficients in Banach spaces. We survey on various aspects of these new developments.
Publications of the Research Institute for Mathematical Sciences, 2003
We show that for every Banach space X the set of 2-homogeneous continuous polynomials whose canon... more We show that for every Banach space X the set of 2-homogeneous continuous polynomials whose canonical extension to X * * attain their norm is a dense subset of the space of all 2-homogeneous continuous polynomials P( 2 X).
A Banach spaceE is known to be Arens regular if every continuous linear mapping fromE toE0 is wea... more A Banach spaceE is known to be Arens regular if every continuous linear mapping fromE toE0 is weakly compact. LetU be an open subset ofE, and letHb(U) denote the algebra of analytic functions onU 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 Fr echet
In [13] Mazet proved the following result. If U is an open subset of a locally convex space E the... more 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 5 U :U^(S(U) such that for any complete locally convex space F and any / e%C{U\F), the space of holomorphic mappings from U to F, there exists a unique linear mapping 7}: ^(U)-* F such that the following diagram commutes;
Journal für die reine und angewandte Mathematik (Crelles Journal), 2000
Page 1. J. reine angew. Math. 557 (2003), 173197 Journal für die reine und angewandte Mathematik... more Page 1. J. reine angew. Math. 557 (2003), 173197 Journal für die reine und angewandte Mathematik ( Walter de Gruyter Berlin Á New York 2003 Bohr's power series theorem and local Banach space theory To the memory of our friend Klaus Floret ...
We obtain general lower and upper estimates for the first and the second Bohr radii of bounded co... more We obtain general lower and upper estimates for the first and the second Bohr radii of bounded complete Reinhardt domains in C n : r 2004 Elsevier Inc. All rights reserved.
THE COINCIDENCE OF T0 AND TC,, FOR SPACES OF HOLOMORPHIC FUNCTIONS ON SOME FRECHET-MONTEL SPACES ... more THE COINCIDENCE OF T0 AND TC,, FOR SPACES OF HOLOMORPHIC FUNCTIONS ON SOME FRECHET-MONTEL SPACES ... By PABLO GALINDO, DOMINGO GARCIA and MANUEL MAESTRE ... Departamento de Andlisis Matemaitico, Universitat de Valencia ... [Received 26 March ...
Journal of Mathematical Analysis and Applications, 2014
ABSTRACT We characterize the Banach spaces Y for which certain subspaces of operators from L1(μ)L... more ABSTRACT We characterize the Banach spaces Y for which certain subspaces of operators from L1(μ)L1(μ) into Y have the Bishop–Phelps–Bollobás property in terms of a geometric property of Y, namely AHSP. This characterization applies to the spaces of compact and weakly compact operators. New examples of Banach spaces Y with AHSP are provided. We also obtain that certain ideals of Asplund operators satisfy the Bishop–Phelps–Bollobás property.
Journal of Mathematical Analysis and Applications, 2014
ABSTRACT In this paper we study some geometrical properties of certain classes of uniform algebra... more ABSTRACT In this paper we study some geometrical properties of certain classes of uniform algebras, in particular the ball algebra image of all uniformly continuous functions on the closed unit ball and holomorphic on the open unit ball of a complex Banach space image. We prove that image has image-numerical index 1 for every image, the lushness and also the AHSP. Moreover, the disk algebra image, and more in general any uniform algebra whose Choquet boundary has no isolated points, is proved to have the polynomial Daugavet property. Most of those properties are extended to the vector valued version image of a uniform algebra image.
Given an entire mapping f ∈ H b (X, X) of bounded type from a Banach space X into X, we denote by... more Given an entire mapping f ∈ H b (X, X) of bounded type from a Banach space X into X, we denote by f the Aron-Berner extension of f to the bidual X * * of X. We show that g • f = g • f if X is symmetrically regular. We also give a counterexample on 1 such that the equality does not hold. We prove that the closure of the numerical range of f is the same as that off .
If X is an Asplund space, then every uniformly continuous function on B X * which is holomorphic ... more If X is an Asplund space, then every uniformly continuous function on B X * which is holomorphic on the open unit ball, can be perturbed by a w * continuous and homogeneous polynomial on X * to obtain a norm attaining function on the dual unit ball. This is a consequence of a version of Bourgain-Stegall's variational principle. We also show that the set of N -homogeneous polynomials between two Banach spaces X and Y whose transposes attain their norms is dense in the corresponding space of N -homogeneous polynomials. In the case when Y is the space of Radon measures on a compact K, this result can be strengthened.
Journal of Mathematical Analysis and Applications, 1987
The holomorphically ultrabornological spaces are introduced. Their relation with other holomorphi... more 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 posed by Nachbin.
Proceedings of the American Mathematical Society, 2007
We prove a multilinear version of Phelps' Lemma: if the zero sets of multilinear forms of norm on... more We prove a multilinear version of Phelps' Lemma: if the zero sets of multilinear forms of norm one are 'close', then so are the multilinear forms.
Each Dirichlet series $D = \sum_{n=1}^{\infty} a_n \frac{1}{n^s}$, with variable $s \in \mathbb{C... more Each Dirichlet series $D = \sum_{n=1}^{\infty} a_n \frac{1}{n^s}$, with variable $s \in \mathbb{C}$ and coefficients $a_n \in \mathbb{C}$, has a so called Bohr strip, the largest strip in $\mathbb{C}$ on which $D$ converges absolutely but not uniformly. The classical Bohr-Bohnenblust-Hille theorem states that the width of the largest possible Bohr strip equals $1/2$. Recently, this deep work of Bohr, Bohnenblust and Hille from the beginning of the last century was revisited by various authors. New methods from different fields of modern analysis (e.g. probability theory, number theory, functional and Fourier analysis) allow to improve the Bohr-Bohnenblust-Hille cycle of ideas, and to extend it to new settings, in particular to Dirichlet series which coefficients in Banach spaces. We survey on various aspects of these new developments.
Publications of the Research Institute for Mathematical Sciences, 2003
We show that for every Banach space X the set of 2-homogeneous continuous polynomials whose canon... more We show that for every Banach space X the set of 2-homogeneous continuous polynomials whose canonical extension to X * * attain their norm is a dense subset of the space of all 2-homogeneous continuous polynomials P( 2 X).
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