Searcharxiv⌕ Search

arXiv subjects

Manuel Maestre

Publications and source records attributed to Manuel Maestre.

18 recordsLinked to original sources
arXiv2026

On a problem of Johnson and Wolfe

In 1979, Johnson and Wolfe proved that norm-attaining operators are dense in $L(C(K),C(S))$ when $K$ and $S$ are compact Hausdorff spaces in the real setting. The corresponding complex case has remained open since then, mainly because the real proof relies on order and sign-decomposition arguments that are no longer available for complex measures. In this paper, we settle the complex case. We prove that, for arbitrary compact Hausdorff spaces $K$ and $S$, the set of norm-attaining operators from the complex space $C(K)$ into the complex space $C(S)$ endowed with the supremum norm is dense in $L(C(K),C(S))$. The proof replaces the real order-theoretic mechanism by a measure-theoretic phase-correction argument, based on polar decompositions, unimodular approximation, and a semicontinuity principle for weighted total variation. This yields a complex defect-reduction procedure which recovers the Johnson-Wolfe density theorem in full generality for complex $C(K)$-spaces.

math.FA↗
arXiv2026

A note on differentials of holomorphic functions

Recently, in arXiv:2304.07149, a bridge was made between the very active area of spaces of Lipschitz real functions on a metric space and holomorphic functions on an open subset of a Banach space. This was done by introducing and studying the space $\mathcal HL_0(B_X)$ of holomorphic Lipschitz functions defined on $B_X$, the open unit ball of the complex Banach space $X$ vanishing at 0. There it was proved that this space is isometrically isomorphic to a subspace of $\mathcal H^\infty(B_X, X^*)$, the space of bounded holomorphic mapping with values in the topological dual of $X$. In that paper it was shown that this subspace was a proper one, except in the one dimensional case. The goal of this note is to give an intrinsic characterization of the elements of that subspace. Moreover, in the case where $X$ additionally has a Schauder basis, it is shown that there is an explicit way to calculate whether and element of $\mathcal H^\infty(B_X, X^*)$ belongs or not to that subspace.

math.FA↗
arXiv2024

Fibers and Gleason parts for the maximal ideal space of $\mathcal A_u(B_{\ell_p})$

In the early nineties, R. M. Aron, B. Cole, T. Gamelin and W.B. Johnson initiated the study of the maximal ideal space (spectrum) of Banach algebras of holomorphic functions defined on the open unit ball of an infinite dimensional complex Banach space. Within this framework, we investigate the fibers and Gleason parts of the spectrum of the algebra of holomorphic and uniformly continuous functions on the unit ball of $\ell_p$ ($1\le p<\infty$). We show that the inherent geometry of these spaces provides a fundamental ingredient for our results. We prove that whenever $p\in\mathbb N$ ($p\ge 2$), the fiber of every $z\in B_{\ell_p}$ contains a set of cardinal $2^{\mathfrak c}$ such that any two elements of this set belong to different Gleason parts. For the case $p=1$, we complete the known description of the fibers, showing that, for each $z\in\overline B_{\ell_1''}\setminus S_{\ell_1}$, the fiber over $z$ is not a singleton. Also, we establish that different fibers over elements in $S_{\ell_1''}$ cannot share Gleason parts.

math.FA↗
arXiv2024

Geometry of homogeneous polynomials in ${\mathbb R}^2$

This work is a thorough and detailed study on the geometry of the unit sphere of certain Banach spaces of homogeneous polynomials in ${\mathbb{R}}^2$. Specifically, we provide a complete description of the unit spheres, identify the extreme points of the unit balls, derive explicit formulas for the corresponding polynomial norms, and describe the techniques required to tackle these questions. To enhance the comprehensiveness of this work, we complement the results and their proofs with suitable diagrams and figures. The new results presented here settle some open questions posed in the past. For the sake of completeness of this work, we briefly discuss previous known results and provide directions of research and applications of our results.

math.FA↗
arXiv2024

On density and Bishop-Phelps-Bollobás type properties for the minimum norm

We study the set $\operatorname{MA}(X,Y)$ of operators between Banach spaces $X$ and $Y$ that attain their minimum norm, and the set $\operatorname{QMA}(X,Y)$ of operators that quasi attain their minimum norm. We characterize the Radon-Nikodym property in terms of operators that attain their minimum norm and obtain some related results about the density of the sets $\operatorname{MA}(X,Y)$ and $\operatorname{QMA}(X,Y)$. We show that every infinite-dimensional Banach space $X$ has an isomorphic space $Y$ such that not every operator from $X$ to $Y$ quasi attains its minimum norm. We introduce and study Bishop-Phelps-Bollobás type properties for the minimum norm, including the ones already considered in the literature, and we exhibit a wide variety of results and examples, as well as exploring the relations between them.

math.FA↗
arXiv2023

Linearization of holomorphic Lipschitz functions

Let $X$ and $Y$ be complex Banach spaces with $B_X$ denoting the open unit ball of $X.$ This paper studies various aspects of the {\em holomorphic Lipschitz space} $\mathcal HL_0(B_X,Y)$, endowed with the Lipschitz norm. This space is the intersection of the spaces, $\operatorname{Lip}_0(B_X,Y)$ of Lipschitz mappings and $\mathcal H^\infty(B_X,Y)$ of bounded holomorphic mappings, from $B_X$ to $Y$. Thanks to the Dixmier-Ng theorem, $\mathcal HL_0(B_X, \mathbb C)$ is indeed a dual space, whose predual $\mathcal G_0(B_X)$ shares linearization properties with both the Lipschitz-free space and Dineen-Mujica predual of $\mathcal H^\infty(B_X)$. We explore the similarities and differences between these spaces, and combine techniques to study the properties of the space of holomorphic Lipschitz functions. In particular, we get that $\mathcal G_0(B_X)$ contains a 1-complemented subspace isometric to $X$ and that $\mathcal G_0(X)$ has the (metric) approximation property whenever $X$ has it. We also analyze when $\mathcal G_0(B_X)$ is a subspace of $\mathcal G_0(B_Y)$, and we obtain an analogous to Godefroy's characterization of functionals with a unique norm preserving extension to the holomorphic Lipschitz context.

math.FA↗
arXiv2021

The spectra of Banach algebras of holomorphic functions on polydisk type domains

R.M. Aron et al. 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, B.J. Cole and T.W. Gamelin showed that $\mathcal{H}^\infty (\ell_2 \cap B_{c_0})$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$ in the sense of an algebra. Motivated by this work, we are interested in a class of open subsets $U$ of a Banach space $X$ for which $\mathcal{H}^\infty (U)$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$. We prove that there exist polydisk type domains $U$ of any infinite dimensional Banach space $X$ with a Schauder basis such that $\mathcal{H}^\infty (U)$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$, which generalizes the result by Cole and Gamelin. Furthermore, we study the analytic and algebraic structure of the spectrum of $\mathcal{H}^\infty (U)$ and show that the Cluster Value Theorem is true for $\mathcal{H}^\infty (U)$.

math.FA↗
arXiv2021

On the compact operators case of the Bishop-Phelps-Bollobás property for numerical radius

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 pass the BPBp-nu for compact operators from subspaces to the whole space and, on the other hand, we prove some strong approximation property of $C_0(L)$ spaces and their duals. Besides, we also show that real Hilbert spaces and isometric preduals of $\ell_1$ have the BPBp-nu for compact operators.

math.FA↗
arXiv2020

Group invariant separating polynomials on a Banach space

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 polynomial $Q$, then $z$ can also be separated from $K$ by a $G$-invariant continuous polynomial $P$. It turns out that this result does not hold when $X$ is a complex Banach space, so we present some additional conditions to get analogous results for the complex case. We also obtain separation theorems under the assumption that $X$ has a Schauder basis which give applications to several classical groups. In this case, we obtain characterizations of points which can be separated by a group invariant polynomial from the closed unit ball.

math.FA↗
arXiv2019

Composition operators on spaces of double Dirichlet series

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 holomorphic functions. Finally, we give a characterization of the superposition operators in $\HC$ and in the spaces $\mathcal{H}^p$.

math.FA↗
arXiv2019

Gleason parts for algebras of holomorphic functions on the ball of $\mathbf{c_0}$

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 homomorphisms $\mathcal M(\mathcal A)$ on $\mathcal A.$ Following remarks on the general situation, we focus on the case $X = c_0.$

math.FA↗
arXiv2017

A non-linear Bishop-Phelps-Bollobás type theorem

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 complex convex dual space of an order continuous absolute normed space. The vector-valued case is also studied.

math.FA↗
arXiv2016

Dirichlet approximation and universal Dirichlet series

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.

math.CV↗
arXiv2016

The Bishop-Phelps-Bollobás property for compact operators

We study the Bishop-Phelps-Bollobás 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, then so do $(C_0(L),Y)$ for every locally compact Hausdorff topological space $L$ and $(X,Y)$ whenever $X^*$ is isometrically isomorphic to $\ell_1$. If $X^*$ has the Radon-Nikodým property and $(\ell_1(X),Y)$ has the BPBp for compact operators, then so does $(L_1(μ,X),Y)$ for every positive measure $μ$; as a consequence, $(L_1(μ,X),Y)$ has the the BPBp for compact operators when $X$ and $Y$ are finite-dimensional or $Y$ is a Hilbert space and $X=c_0$ or $X=L_p(ν)$ for any positive measure $ν$ and $1< p< \infty$. For $1\leqslant p <\infty$, if $(X,\ell_p(Y))$ has the BPBp for compact operators, then so does $(X,L_p(μ,Y))$ for every positive measure $μ$ such that $L_1(μ)$ is infinite-dimensional. If $(X,Y)$ has the BPBp for compact operators, then so do $(X,L_\infty(μ,Y))$ for every $σ$-finite positive measure $μ$ and $(X,C(K,Y))$ for every compact Hausdorff topological space $K$.

math.FA↗
arXiv2014

The Dirichlet-Bohr radius

Denote by $Ω(n)$ 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 $$ \sum_{n \leq x} |a_n| r^{Ω(n)} \leq \sup_{t\in \mathbb{R}} \big|\sum_{n \leq x} a_n n^{-it}\big|\,. $$ We prove that the asymptotically correct order of $L(x)$ is $ (\log x)^{1/4}x^{-1/8} $. Following Bohr's vision our proof links the estimation of $L(x)$ with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows to translate various results on Bohr radii in a systematic way into results on Dirichlet-Bohr radii, and vice versa.

math.NT↗
arXiv2014

Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables

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 $D \in \mathcal{H}_\infty$. We study the problem of describing such sequences $(b_n)$ in terms of the asymptotic decay of the subsequence $(b_{p_j})$, where $p_j$ denotes the $j$th prime number. Given a multiplicative sequence $b=(b_n)$ we prove (among other results): $b$ is an $\ell_1$-multiplier for $\mathcal{H}_\infty$ provided $|b_{p_j}| < 1$ for all $j$ and $\overline{\lim}_n \frac{1}{\log n} \sum_{j=1}^n b_{p_j}^{*2} < 1$, and conversely, if $b$ is an $\ell_1$-multiplier for $\mathcal{H}_\infty$, then $|b_{p_j}| < 1$ for all $j$ and $\overline{\lim}_n \frac{1}{\log n} \sum_{j=1}^n b_{p_j}^{*2} \leq 1$ (here $b^*$ stands for the decreasing rearrangement of $b$). Following an ingenious idea of Harald Bohr it turns out that this problem is intimately related with the question of characterizing those sequences $z$ in the infinite dimensional polydisk $\mathbb{D}^\infty$ (the open unit ball of $\ell_\infty$) for which every bounded and holomorphic function $f$ on $\mathbb{D}^\infty$ has an absolutely convergent monomial series expansion $\sum_α \frac{\partial_αf(0)}{α!} z^α$. Moreover, we study analogous problems in Hardy spaces of Dirichlet series and Hardy spaces of functions on the infinite dimensional polytorus $\mathbb{T}^\infty$.

math.FA↗
arXiv2012

Monomial expansions of $H_{p}$--functions in infinitely many variables

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 $f \in H_\infty(\mathbb{D}^\infty)$ sums up to $f(z)$. Bohr, Bohnenblust and Hille, showed that it contains $\ell_{2} \cap \mathbb{D}^\infty$, but does not contain any of the slices $\ell_{2+\varepsilon} \cap \mathbb{D}^\infty$. This was done in the context of Dirichlet series and our article is very much inspired by recent deep developments in this direction. Our main contribution shows that $z \in \mon H_\infty(\mathbb{D}^\infty)$ whenever $\bar{\lim} \big(\frac{1}{\log n} \sum_{j=1}^{n} z^{* 2}_{j} \big)^{1/2} < 1/\sqrt{2}$, and conversely $\bar{\lim} \big(\frac{1}{\log n} \sum_{j=1}^{n} z^{* 2}_{j} \big)^{1/2} \leq 1$ for each $z \in \mon H_\infty(\mathbb{D}^\infty)$. The Banach space $H_\infty(\mathbb{D}^\infty)$ can be identified with the Hardy space $H_\infty(\mathbb{T}^\infty)$; this motivates a study of sets of monomial convergence of $H_p$-functions on $\mathbb{T}^\infty$ (consisting of all $z$'s in $\mathbb{D}^{\infty}$ for which the series $\sum \hat{f}(α) z^α$ converges). We show that $\mon H_\infty(\mathbb{T}^\infty) = \mon H_\infty(\mathbb{D}^\infty)$ and $\mon H_{p}(\mathbb{T}^\infty) = \ell_{2} \cap \mathbb{D}^\infty$ for $1 \leq p < \infty$ and give a representation of $H_{p}(\mathbb{T}^\infty)$ in terms of holomorphic functions on $\mathbb{D}^{\infty}$. This links our circle of ideas with well-known results due to Cole and Gamelin.

math.FA↗