# Manuel MaestreUniversity of Valencia | UV · Departament of Mathematical Analysis

Manuel Maestre

Professor, Phd. Dr.

## About

155

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Introduction

Additional affiliations

August 2010 - August 2011

September 1979 - present

## Publications

Publications (155)

In this survey, we provide an overview from 2008 to 2021 about the Bishop–Phelps–Bollobás theorem.KeywordsNorm attaining operatorsBishop–Phelps theoremBishop–Phelps–Bollobás property

Aron et al. (Math Ann 353:293–303, 2012) proved that the Cluster Value Theorem in the infinite dimensional Banach space setting holds for the Banach algebra \(\mathcal {H}^\infty (B_{c_0})\). On the other hand, Cole and Gamelin (Proc Lond Math Soc 53:112–142, 1986) showed that \(\mathcal {H}^\infty (\ell _2 \cap B_{c_0})\) is isometrically isomorph...

We study the Bishop–Phelps–Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that \(C_0(L)\) spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space L. To this end, on the one hand, we provide some techniques allowing to pa...

A new estimate for the Bohr radius of the family of holomorphic functions in the n-dimensional polydisk is provided. This estimate, obtained via a new approach, is sharper than those that are known up to date.

We study the Bishop-Phelps-Bollob\'as property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that $C_0(L)$ spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space $L$. To this end, on the one hand, we provide some techniques allowing to...

R.M. Aron et al. [3] presented that the Cluster Value Theorem in the infinite dimensional Banach space setting holds for the Banach algebra H ∞ (B c0). On the other hand, B.J. Cole and T.W. Gamelin [15] showed that H ∞ (2 ∩ B c0) is isometrically isomorphic to H ∞ (B c0) in the sense of an algebra. Motivated by this work, we are interested in a cla...

We introduce and investigate the mth polarization constant of a Banach space X for the numerical radius. We first show the difference between this constant and the original mth polarization constant associated with the norm by proving that the new constant is minimal if and only if X is strictly convex, and that there exists a Banach space which do...

We study the group invariant continuous polynomials on a Banach space $X$ that separate a given set $K$ in $X$ and a point $z$ outside $K$. We show that if $X$ is a real Banach space, $G$ is a compact group of $\mathcal{L} (X)$, $K$ is a $G$-invariant set in $X$, and $z$ is a point outside $K$ that can be separated from $K$ by a continuous polynomi...

Let H∞(Bc0) be the algebra of all bounded holomorphic functions on the open unit ball of c0 and M(H∞(Bc0)) the spectrum of H∞(Bc0). We prove that for any point z in the closed unit ball of ℓ∞ there exists an analytic injection of the open ball Bℓ∞ into the fiber of z in M(H∞(Bc0)), which is an isometry from the Gleason metric of Bℓ∞ to the Gleason...

We study composition operators on spaces of double Dirichlet series, focusing our interest on the characterization of the composition operators of the space of bounded double Dirichlet series \({\mathcal {H}}^\infty ({\mathbb {C}}_+^2)\). We also show how the composition operators of this space of Dirichlet series are related to the composition ope...

For a complex Banach space X with open unit ball BX, consider the Banach algebras H∞(BX) of bounded scalar-valued holomorphic functions and the subalgebra Au(BX) of uniformly continuous functions on BX. Denoting either algebra by A, we study the Gleason parts of the set of scalar-valued homomorphisms M(A) on A. Following remarks on the general situ...

Cambridge Core - Abstract Analysis - Dirichlet Series and Holomorphic Functions in High Dimensions - by Andreas Defant

We study composition operators on spaces of double Dirichlet series, focusing our interest on the characterization of the composition operators of the space of bounded double Dirichlet series $\HCdos$. We also show how the composition operators of this space of Dirichlet series are related to the composition operators of the corresponding spaces of...

For a complex Banach space $X$ with open unit ball $B_X,$ consider the Banach algebras $\mathcal H^\infty(B_X)$ of bounded scalar-valued holomorphic functions and the subalgebra $\mathcal A_u(B_X)$ of uniformly continuous functions on $B_X.$ Denoting either algebra by $\mathcal A,$ we study the Gleason parts of the set of scalar-valued homomorphism...

In the study of the spectra of algebras of holomorphic functions on a Banach space E , the bidual E ″ has a central role, and the spectrum is often shown to be locally homeomorphic to E ″. In this paper we consider the problem of spectra of subalgebras invariant under the action of a group (functions f such that f ○ g = f ). It is natural to attemp...

In this paper we study spaces of multiple Dirichlet series and their properties. We set the ground of the theory of multiple Dirichlet series and define the spaces H∞(C+k), k∈N, of convergent and bounded multiple Dirichlet series on C+k. We give a representation for these Banach spaces and prove that they are all isometrically isomorphic, independe...

Giving a partial answer to a conjecture formulated by Aron, Boyd, Ryan and Zalduendo, we show that if a real Banach space X is not linearly and continuously injected into a Hilbert space, then for any 2-homogeneous continuous polynomial P on X, its zero-set is not separable. For this class of spaces, we also prove that, if P is semidefinite, then i...

We study when the spaces of general Dirichlet series bounded on a half plane are Banach spaces, and show that some of those classes are isometrically isomorphic between themselves. In a precise way, let {λn} be a strictly increasing sequence of positive real numbers such that limn→∞λn=∞. We denote by H∞(λn) the complex normed space of all Dirichle...

Given a proper holomorphic mapping \(g:\varOmega \subseteq {\mathbb {C}}^{n}\longrightarrow \varOmega ' \subseteq {\mathbb {C}}^{n}\) and an algebra of holomorphic functions \({\mathcal {B}}\) (e.g. \({\mathscr {P}}(K)\) where \(K\subset \varOmega \) is a compact set, \({\mathcal {H}}(U)\), A(U) or \({\mathcal {H}}^{\infty }(U)\) where U is an open...

Let $\mathcal{H}_\infty$ be the set of all ordinary Dirichlet series
$D=\sum_n a_n n^{-s}$ representing bounded holomorphic functions on the right
half plane. A multiplicative sequence $(b_n)$ of complex numbers is said to be
an $\ell_1$-multiplier for $\mathcal{H}_\infty$ whenever $\sum_n |a_n b_n| <
\infty$ for every $D \in \mathcal{H}_\infty$. W...

The main aim of this paper is to prove a Bishop-Phelps-Bollobás type theorem on the unital uniform algebra A w∗u (B X∗ ) consisting of all w∗-uniformly continuous functions on the closed unit ball B X∗ which are holomorphic on the interior of B X∗ . We show that this result holds for A w∗u (B X∗ ) if X∗ is uniformly convex or X∗ is the uniformly co...

We study the existence of separation theorems by polynomials that are invariant under a group action. We show that if G is a finite subgroup of \(\textit{GL}(n,{\mathbb {C}})\), K is a set in \({\mathbb {C}}^{n}\) that is invariant under the action of G and z is a point in \({\mathbb {C}}^{n}\setminus K\) that can be separated from K by a polynomia...

We study the existence of infinite-dimensional vector spaces in the sets of norm-attaining operators, multilinear forms, and polynomials. Our main result is that, for every set of permutations P of the set [1,...,n], there exists a closed infinite-dimensional Banach subspace of the space of n-linear forms on ℓ1 such that, for all nonzero elements B...

Let BX be the open unit ball of a complex Banach space X, and let H∞(BX) and Au(BX) be, respectively, the algebra of bounded holomorphic functions on BX and the subalgebra of uniformly continuous holomorphic functions on BX: In this paper we study the analytic structure of fibers in the spectrum of these two algebras. For the case of H∞ (BX), we pr...

We study the Bishop-Phelps-Bollobás property and the Bishop-Phelps-Bollobás property for numerical radius. Our main aim is to extend some known results about norm or numerical radius attaining operators to multilinear and polynomial cases. We characterize the pair $(\ell_1(X), Y)$ to have the BPBp for bilinear forms and prove that on $L_1 (\mu)$ th...

We characterize the uniform limits of Dirichlet polynomials on a right half plane. In the Dirichlet setting, we find approximation results, with respect to the Euclidean distance and {to} the chordal one as well, analogous to classical results of Runge, Mergelyan and Vitushkin. We also strengthen the notion of universal Dirichlet series.

We study the Bishop-Phelps-Bollob\'as property (BPBp for short) for compact operators. We present some abstract techniques which allows to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$ has the BPBp for compact operators,...

Given two balanced compact subsets K and L of two Banach spaces X and Y respectively such that every continuous m-homogeneous polynomial on \(X^{**}\) and on \(Y^{**}\) is approximable, for all \(m\in \mathbb {N}\), we characterize when the algebras of holomorphic germs \(\mathcal {H}(K)\) and \(\mathcal {H}(L)\) are topologically algebra isomorphi...

We exhibit a new class of Banach spaces Y such that the pair has the Bishop-Phelps-Bollobás property for operators. This class contains uniformly convex Banach spaces and spaces with the property β of Lindenstrauss. We also provide new examples of spaces in this class.

In this paper we give a new characterization of when a Banach space E has the Schur property in terms of the disk algebra. We prove that E has the Schur property if and only if .

We study the behavior of holomorphic mappings on p-compact sets in Banach spaces. We show that the image of a p-compact set by an entire mapping is a p-compact set. Some results related to the localization of p-compact sets in the predual of homogeneous polynomials are also obtained. Finally, the “size” of p-compactness of the image of the unit bal...

Denote by $\Omega(n)$ the number of prime divisors of $n \in \mathbb{N}$
(counted with multiplicities). For $x\in \mathbb{N}$ define the Dirichlet-Bohr
radius $L(x)$ to be the best $r>0$ such that for every finite Dirichlet
polynomial $\sum_{n \leq x} a_n n^{-s}$ we have $$ \sum_{n \leq x} |a_n|
r^{\Omega(n)} \leq \sup_{t\in \mathbb{R}} \big|\sum_{...

Our goal is to study the Bishop-Phelps-Bollobas property for operators from c(0) into a Banach space. We first characterize those Banach spaces Y for which the Bishop-Phelps-Bollobas property holds for (l(infinity)(3), Y). Examples of spaces satisfying this condition are provided.

Denote by Ω(n) the number of prime divisors of n ∈ N (counted with multi-plicities). For x ∈ N define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial n≤x a n n −s we have n≤x |a n |r Ω(n) ≤ sup t ∈R n≤x a n n −i t. We prove that the asymptotically correct order of L(x) is (log x) 1/4 x −1/8. Follo...

For a sigma-finite measure mu and a Banach space Y we study the Bishop-Phelps-Bollobas property (BPBP) for bilinear forms on L-1(mu) X Y, that is, a (continuous) bilinear form on L-1(mu) X Y almost attaining its norm at (f(0), y(0)) can be approximated by bilinear forms attaining their norms at unit vectors close to (f(0), y(0)). In case that Y is...

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 t...

For two complex Banach spaces $X$ and $Y$, in this paper we study the generalized spectrum $\mathcal{M}_b(X,Y)$ of all non-zero algebra homomorphisms from $\mathcal{H}_b(X)$, the algebra of all bounded type entire functions on $X$, into $\mathcal{H}_b(Y)$. We endow $\mathcal{M}_b(X,Y)$ with a structure of Riemann domain over $\mathcal{L}(X^*,Y^*)$...

We study norm attaining properties of the Arens extensions of multilinear forms defined on Banach spaces. Among other related results, we construct a multilinear form on ℓ1 with the property that only some fixed Arens extensions determined a priori attain their norms. We also study when multilinear forms can be approximated by ones with the propert...

We study the cluster value theorem for Hb(X), the Fréchet algebra of holomorphic functions bounded on bounded sets of X. We also describe the (size of) fibers of the spectrum of Hb(X). Our results are rather complete whenever X has an unconditional shrinking basis and for X = ℓ1. As a byproduct, we obtain results on the spectrum of the algebra of a...

In this survey we report on very recent results about some non-linear geometrical properties of many classes of real and complex Banach spaces and uniform algebras, including the ball algebra \(\fancyscript{A}_u(B_X)\) of all uniformly continuous functions on the closed unit ball and holomorphic on the open unit ball of a complex Banach space \(X\)...

We estimate the polynomial numerical indices of the spaces C(K) and L1(μ).

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 th...

In this paper we provide versions of the Bishop-Phelps-Bollobás Theorem for bilinear forms. Indeed we prove the first positive result of this kind by assuming uniform convexity on the Banach spaces. A characterization of the Banach space Y satisfying a version of the Bishop-Phelps-Bollobás Theorem for bilinear forms on 1 × Y is also obtained. As a...

In this paper, we prove that the Bishop-Phelps-Bollobas Theorem holds for hermitian forms on a complex Hilbert space and for symmetric bilinear forms on a real or complex Hilbert space. The Bishop-Phelps-Bollobas Theorem for operators in the Schatten-von Neumann classes is also obtained.

Motivated by the scalar case we study Bohr radii of the NN-dimensional polydisc DNDN for holomorphic functions defined on DNDN with values in Banach spaces.

We study the spectra of algebras of holomorphic functions with prescribed radii of boundedness, and use these results to study the τω and τδ spectra of H(U), where U is an open subset of a non-separable Banach space. We construct τδ continuous characters on H(U) which are not evaluations at points of U. We also discuss subsets of ℓ∞ which are bound...

Each bounded holomorphic function on the infinite dimensional polydisk
$\mathbb{D}^\infty$, $f \in H_\infty(\mathbb{D}^\infty)$, defines a formal
monomial series expansion that in general does not converge to $f$. The set
$\mon H_\infty(\mathbb{D}^\infty)$ contains all $ z $'s in which the monomial
series expansion of each function $f \in H_\infty(...

We investigate uniform algebras of bounded analytic functions on the unit ball of a complex Banach space. We prove several
cluster value theorems, relating cluster sets of a function to its range on the fibers of the spectrum of the algebra. These
lead to weak versions of the corona theorem for ℓ
2 and for c
0. In the case of the open unit ball of...

Roughly speaking, a regular surface in \(\mathbb{R}^3\) is a two-dimensional set of points, in the sense that it can be locally described by two parameters (the local coordinates) and with the property that it is smooth enough (that is, there are no vertices, edges, or self-intersections) to guarantee the existence of a tangent plane to the surface...

In studying the motion of a particle along an arc it is convenient to consider the arc as the image of a vector-valued mapping \(\gamma: [a,b] \rightarrow \mathbb{R}^3\) defined on an interval of the real line and realize γ(t) as the position of the particle at time t. This viewpoint is also convenient in analyzing the behavior of a vector field al...

We intend to study the integration of a differential k-form over a regular k-surface of class C
1 in \(\mathbb{R}^n\). To begin with, in Sect. 7.1, we undertake the integration over a portion of the surface that is contained in a coordinate neighborhood. Where possible, we will express the obtained results in terms of integration of vector fields....

One of the objectives of this book is to obtain a rigorous proof of a version of Green’s formula for compact subsets of \(\mathbb{R}^2\) whose topological boundary is a regular curve of class C
2. These sets are typical examples of what we will call regular 2-surfaces with boundary in \(\mathbb{R}^2\). The analogous three-dimensional example would...

In this chapter we concentrate on aspects of vector calculus. A common physical application of this theory is the fluid flow problem of calculating the amount of fluid passing through a permeable surface. The abstract generalization of this leads us to the flux of a vector field through a regular 2-surface in \(\mathbb{R}^3\). More precisely, let t...

We know from Chap. 4 that in order to evaluate the flux of a vector field across a regular surface S, we need to choose a unit normal vector at each point of S in such a way that the resulting vector field is continuous. For instance, if we submerge a permeable sphere into a fluid and we select the field of unit normal outward vectors on the sphere...

Let ω be a differential form of degree k - 1 and class C
1 in a neighborhood of a compact regular k-surface with boundary M of class C
2. The general Stokes’s theorem gives a relationship between the integral of ω over the boundary of M and the integral of the exterior differential dω over M. It can be viewed as a generalization of Green’s theorem...

The purpose of this book is to explain in a rigorous way Stokes’s theorem and to facilitate the student’s use of this theorem in applications. Neither of these aims can be achieved without first agreeing on the notation and necessary background concepts of vector calculus, and therein lies the motivation for our introductory chapter.

The aim of this book is to facilitate the use of Stokes' Theorem in applications. The text takes a differential geometric point of view and provides for the student a bridge between pure and applied mathematics by carefully building a formal rigorous development of the topic and following this through to concrete applications in two and three varia...

We survey the most relevant recent developments on the research of the spectra of algebras of analytic functions. We concentrate mainly on three algebras, the Banach algebra H ∞ (B) of all bounded holomorphic functions on the unit ball B of a complex Banach space X, the Banach algebra of the ball A u (B), and the Fréchet algebra H b (X) of all enti...

We show that the Bishop–Phelps–Bollobás theorem holds for all bounded operators from L1(μ)L1(μ) into L∞[0,1]L∞[0,1], where μ is a σ-finite measure.

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...

We investigate certain envelopes of open sets in dual Banach spaces which are related to extending holomorphic functions. We give a variety of examples of absolutely convex sets showing that the extension is in many cases not possible. We also establish connections to the study of iterated weak* sequential closures of convex sets in the dual of sep...

In this article we study the interplay of the theory of classical Dirichlet series in one complex variable with recent development on monomial expansions of holomorphic functions in infinitely many variables. For a given Dirichlet series we obtain new strips of convergence in the complex plane related to Bohr’s classical strips of uniform but non a...

The aim of this paper is to show that there exist infinite dimensional Banach spaces of functions that, except for 0, satisfy properties that apparently should be destroyed by the linear combination of two of them. Three of these spaces are: a Banach space of differentiable functions on ℝn failing the Denjoy-Clarkson property; a Banach space of non...

Following Sinha and Karn [9], a relatively compact subset K of a Banach space E is said to be p-compact if for some sequence (x
n
) ∈ l
p
(E), K ⊂ {Σ
n
a
x
x
n
| (a
n
) ∈ B
ℓ′
p
}. In [4], Delgado, Oja, Piñeiro, and Serrano investigated the p-approximation property, in which one only requires finite rank approximation of the identity on p-co...

If X is an Asplund space, then every uniformly continuous function on BX* 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 se...

Let ℱ(R) be a set of holomorphic functions on a Reinhardt domain R in a Banach sequence space (as e.g. all holomorphic functions or all m-homogeneous polynomials on the open unit ball of ). We give a systematic study of the sets dom ℱ(R) of all z ∈ R for which the monomial expansion of every ∈ ℱ(R) converges. Our results are based on and improve th...

To the memory of Goyo Sevilla, a good, honest man. Abstract. In this paper we give general conditions on a countable family V of weights on an unbounded open set U in a complex Banach space X such that the weighted space HV (U) of holomorphic functions on U has a Fréchet algebra structure. For that kind of weights it is shown that the spectrum of H...

Let H be a (real or complex) Hilbert space. Using spectral theory and properties of the Schatten–Von Neumann operators, we prove
that every symmetric tensor of unit norm in H [^(Ä)] s,psH{H \hat{\otimes} _{s,\pi _{s}}H} is an infinite absolute convex combination of points of the form xÄx{x\otimes x} with x in the unit sphere of the Hilbert space. W...

We study two-dimensional Banach spaces with polynomial numerical indices equal to zero.

A full-wave analysis method for waveguide filters based on dielectric loaded resonators is proposed in this work. For such purpose, a state-space integral-equation formulation has been developed, and the efficient numerical evaluation of all matrices and singular integrals related to the method has been detailed. The novel technique has been first...

We study two-dimensional Banach spaces with polynomial numerical indices equal to zero.

Let X be a separable Banach space. We provide an explicit construction of a sequence in X that tends to 1 in norm but which is weakly dense. Our interest in the result stated in the Abstract was motivated by two theorems. First, in their work on hypercyclic operators, K. Chan and R. Sanders proved the following: Theorem 1. (Chan and Sanders (3)) Fo...

Given an entire mapping $f\in \mathcal{H}_b(X,X)$ of bounded type from a Banach space $X$ into $X$, we denote by $\overline{f}$ the Aron-Berner extension of $f$ to the bidual $X^{\ast\ast}$ of $X$. We show that $\overline{g\circ f} = \overline{g}\circ \overline{f}$ for all $f, g\in \mathcal{H}_b(X,X)$ if $X$ is symmetrically regular. We also give a...

We prove the Bishop–Phelps–Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop–Phelps–Bollobás theorem holds for operators from ℓ1ℓ1 into Y. Several examples of classes of such spaces are provid...