arXiv subjects

Verónica Dimant

Publications and source records attributed to Verónica Dimant.

At least 19 recordsLinked to original sources

arXiv2026

Composition operators for holomorphic Lipschitz functions

We study composition operators on spaces of holomorphic Lipschitz functions defined on the open unit ball of a complex Banach space. Our approach is based on the linearization of the symbol through the holomorphic Lipschitz-free spaces, which allow composition operators to be realized as adjoints of linear operators. For spaces with the bounded approximation property, we characterize composition operators between spaces of holomorphic Lipschitz functions vanishing at the origin and describe when composition operators are onto isomorphisms. We further investigate compactness and weak compactness properties of composition operators. In the finite-dimensional setting, compactness and weak compactness are shown to coincide, and a complete characterization is obtained in terms of the symbol. Finally, we analyze the asymptotic behavior of the iterates of composition operators, proving convergence to zero whenever the supremum norm of the symbol is less than one, and we extend several results to the case not vanishing at 0.

math.FA↗

arXiv2025

New insights into Gleason parts for an algebra of holomorphic functions

We study the structure of the spectrum of the algebra of uniformly continuous holomorphic functions on the unit ball of $\ell_p$. Our main focus is the relationship between \emph{Gleason parts} and \emph{fibers}. For every $z \in B_{\ell_p}$ with $1 < p < \infty$, we prove that the fiber over $z$ contains $2^{\mathfrak{c}}$ distinct Gleason parts. We also investigate some of the properties of these Gleason parts and show the existence of many strong boundary points in certain fibers. We then examine the case $p = 1$, where similar results on the abundance of Gleason parts within the fibers hold, although the arguments required are more involved. Our results extend and complete earlier work on the subject, providing answers to previously posed questions.

math.CV↗

arXiv2025

The $p$-Operator Approximation Property

We study a notion analogous to the $p$-Approximation Property ($p$-AP) for Banach spaces, within the noncommutative context of operator spaces. Referred to as the $p$-Operator Approximation Property ($p$-OAP), this concept is linked to the ideal of operator $p$-compact mappings. We present several equivalent characterizations based on the density of finite-rank mappings within specific spaces for different topologies, and also one in terms of a slice mapping property. Additionally, we investigate how this property transfers from the dual or bidual to the original space. As an application, the $p$-OAP for the reduced $C^*$-algebra of a discrete group implies that operator $p$-compact Herz-Schur multipliers can be approximated in $\mbox{cb}$-norm by finitely supported multipliers.

math.FA↗

arXiv2025

Revisiting Operator $p$-Compact Mappings

We continue our study of the mapping ideal of operator $p$-compact maps, previously introduced by the authors. Our approach embraces a more geometric perspective, delving into the interplay between operator $p$-compact mappings and matrix sets, specifically we provide a quantitative notion of operator $p$-compactness for the latter. In particular, we consider operator $p$-compactness in the bidual and its relation with this property in the original space. Also, we deepen our understanding of the connections between these mapping ideals and other significant ones (e.g., completely $p$-summing, completely $p$-nuclear).

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

Linearizing holomorphic functions on operator spaces

We introduce a notion of completely bounded holomorphic functions defined on the open unit ball of an operator space. We endow the set of these functions with an operator space structure, and in the scalar-valued case we identify an operator space predual for it which is a noncommutative version of Mujica's predual for the space of bounded holomorphic functions and satisfies similar properties. In particular, our predual is a free holomorphic operator space in the sense that it satisfies a linearization property for vector-valued completely bounded holomorphic functions. Additionally, several different operator space approximation properties transfer between the predual and the domain.

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↗

arXiv2023

Operator space tensor norms

The use of a tensor product perspective has enriched functional analysis and other important areas of mathematics and physics. The context of operator spaces is clearly no exception. The aim of this manuscript is to kick off the development of a systematic theory of tensor products and tensor norms for operator spaces and its interplay with their associated mapping ideals. Based on the theory of tensor products in Banach spaces, we provide the corresponding natural definitions in the operator space framework. The theory is not a mere translation of what is known in the classical setting and new insights, techniques, ideas or hypotheses are required in many cases. As a consequence, notable differences in the theory appear when compared to the classical one.

math.OA↗

arXiv2022

Homomorphisms on algebras of analytic functions on non-symmetrically regular spaces

We study homomorphisms on the algebra of analytic functions of bounded type on a Banach space. When the domain space lacks symmetric regularity, we show that in every fiber of the spectrum there are evaluations (in higher duals) which do not coincide with evaluations in the second dual. We also consider the commutativity of convolutions between evaluations. We show that in some Banach spaces $X$ (for example, $X=\ell_1$) the only evaluations that commute with every other evaluation in $X''$ are those in $X$. Finally, we establish conditions ensuring the symmetry of the canonical extension of a symmetric multilinear operator (on a non-symmetrically regular space) and present some applications.

math.FA↗

arXiv2021

A look into homomorphisms between uniform algebras over a Hilbert space

We study the vector-valued spectrum $\mathcal{M}_{u,\infty}(B_{\ell_2},B_{\ell_2})$ which is the set of nonzero algebra homomorphisms from $\mathcal{A}_u(B_{\ell_2})$ (the algebra of uniformly continuous holomorphic functions on $B_{\ell_2}$) to $\mathcal {H}^\infty(B_{\ell_2})$ (the algebra of bounded holomorphic functions on $B_{\ell_2}$). This set is naturally projected onto the closed unit ball of $\mathcal {H}^\infty(B_{\ell_2}, \ell_2)$ giving rise to an associated fibering. Extending the classical notion of cluster sets introduced by I. J. Schark (1961) to the vector-valued spectrum we define vector-valued cluster sets. The aim of the article is to look at the relationship between fibers and cluster sets obtaining results regarding the existence of analytic balls into these sets.

math.FA↗

arXiv2020

The polarization constant of finite dimensional complex spaces is one

The polarization constant of a Banach space $X$ is defined as $$\mathbf c(X):= \limsup\limits_{k\rightarrow \infty} \mathbf c(k, X)^\frac{1}{k},$$ where $\mathbf c(k, X)$ stands for the best constant $C>0$ such that $ \Vert \overset{\vee}{P} \Vert \leq C \Vert P \Vert$ for every $k$-homogeneous polynomial $P \in \mathcal P(^kX)$. We show that if $X$ is a finite dimensional complex space then $\mathbf c(X)=1$. We derive some consequences of this fact regarding the convergence of analytic functions on such spaces.The result is no longer true in the real setting. Here we relate this constant with the so-called Bochnak's complexification procedure. We also study some other properties connected with polarization. Namely, we provide necessary conditions related with the geometry of $X$ for $\mathbf c(2,X)=1$ to hold. Additionally we link polarization's constants with certain estimates of the nuclear norm of the product of polynomials.

math.FA↗

arXiv2019

Homomorphisms between algebras of holomorphic functions on the infinite polydisk

We study the vector-valued spectrum $\mathcal{M}_\infty(B_{c_0},B_{c_0})$, that is, the set of non null algebra homomorphisms from $\mathcal H^\infty(B_{c_0})$ to $\mathcal H^\infty(B_{c_0})$ which is naturally projected onto the closed unit ball of $\mathcal H^\infty(B_{c_0}, \ell_\infty)$, likewise the scalar-valued spectrum $\mathcal M_\infty(B_{c_0})$ which is projected over $\bar{B}_{\ell_\infty}$. Our itinerary begins in the scalar-valued spectrum $\mathcal{M}_\infty(B_{c_0})$: by expanding a result by Cole, Gamelin and Johnson (1992) we prove that on each fiber there are $2^c$ disjoint analytic Gleason isometric copies of $B_{\ell_\infty}$. For the vector-valued case, building on the previous result we obtain $2^c$ disjoint analytic Gleason isometric copies of $B_{\mathcal{H}^\infty(B_{c_0},\ell_\infty)}$ on each fiber. We also take a look at the relationship between fibers and Gleason parts for both vector-valued spectra $\mathcal{M}_{u,\infty}(B_{c_0},B_{c_0})$ and $\mathcal{M}_\infty(B_{c_0},B_{c_0})$.

math.FA↗

arXiv2019

A fibered description of the vector-valued spectrum

For Banach spaces $X$ and $Y$ we study the vector-valued spectrum $\mathcal M_\infty(B_X,B_Y)$, that is the set of non null algebra homomorphisms from $\mathcal H^\infty(B_X)$ to $\mathcal H^\infty(B_Y)$, which is naturally projected onto the closed unit ball of $\mathcal H^\infty(B_Y, X^{**})$. The aim of this article is to describe the fibers defined by this projection, searching for analytic balls and considering Gleason parts.

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↗

arXiv2018

Operator $p$-compact mappings

We introduce the class of operator $p$-compact mappings and completely right $p$-nuclear operators, which are natural extensions to the operator space framework of their corresponding Banach operator ideals. We relate these two classes, define natural operator space structures and study several properties of these ideals. We show that the class of operator $\infty$-compact mappings in fact coincides with a notion already introduced by Webster in the nineties (in a very different language). This allows us to provide an operator space structure to Webster's class.

math.FA↗

arXiv2017

Diagonal Multilinear Operators on Köthe Sequence Spaces

We analyze the interplay between maximal/minimal/adjoint ideals of multilinear operators (between sequence spaces) and their associated Köthe sequence spaces. We establish relationships with spaces of multipliers and apply these results to describe diagonal multilinear operators from Lorentz sequence spaces. We also define and study some properties of the ideal of $(E;p)$-summing multilinear mappings, a natural extension of the linear ideal of absolutely $(E;p)$-summing operators.

math.FA↗

arXiv2016

Ideal structures in vector-valued polynomial spaces

This paper is concerned with the study of geometric structures in spaces of polynomials. More precisely, we discuss for $E$ and $F$ Banach spaces, whether the class of weakly continuous on bounded sets $n$-homogeneous polynomials, $\mathcal P_w(^n E, F)$, is an HB-subspace or an $M(1,C)$-ideal in the space of continuous $n$-homogeneous polynomials, $\mathcal P(^n E, F)$. We establish sufficient conditions under which the problem can be positively solved. Some examples are given. We also study when some ideal structures pass from $\mathcal P_w(^n E, F)$ as an ideal in $\mathcal P(^n E, F)$ to the range space $F$ as an ideal in its bidual $F^{**}$.

math.FA↗

arXiv2015

Bilinear Ideals in Operator Spaces

We introduce a concept of bilinear ideal of jointly completely bounded mappings between operator spaces. In particular, we study the bilinear ideals $\mathcal{N}$ of completely nuclear, $\mathcal{I }$ of completely integral, $\mathcal{E}$ of completely extendible bilinear mappings, $\mathcal{MB}$ multiplicatively bounded and its symmetrization $\mathcal{SMB}$. We prove some basic properties of them, one of which is the fact that $\mathcal{I}$ is naturally identified with the ideal of (linear) completely integral mappings on the injective operator space tensor product.

math.OA↗