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José Bonet

Publications and source records attributed to José Bonet.

At least 37 records · Page 2Linked to original sources
arXiv2020

Every separable complex Fréchet space with a continuous norm is isomorphic to a space of holomorphic functions

Extending a result of Mashreghi and Ransford, we prove that every complex separable infinite dimensional Fréchet space with a continuous norm is isomorphic to a space continuously included in a space of holomorphic functions on the unit disc or the complex plane, which contains the polynomials as a dense subspace. As a consequence examples of nuclear Fréchet spaces of holomorphic functions without the bounded approximation exist.

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arXiv2019

The spectrum of composition operators induced by a rotation in the space of all analytic functions on the disc

A characterization of those points in the unit disc which belong to the spectrum of a composition operator $C_φ$, defined by a rotation $φ(z)=rz$ with $|r|=1$, on the space $H_0(\mathbb{D})$ of all analytic functions on the unit disc which vanish at $0$, is given. Examples show that the point $1$ may or may not belong to the spectrum of $C_φ$, and this is related to Diophantine approximation. Our results complement recent work by Arendt, Celariès and Chalendar.

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arXiv2019

Order spectrum of the Cesàro operator in Banach lattice sequence spaces

The discrete Cesàro operator $ C $ acts continuously in various classical Banach sequence spaces within $ \mathbb{C}^{\mathbb{N}}.$ For the coordinatewise order, many such sequence spaces $ X $ are also complex Banach lattices (eg. $c_0, \ell^p $ for $ 1 < p \leq \infty , $ and $ ces (p)$ for $ p \in \{ 0 \} \cup ( 1, \infty )).$ In such Banach lattice sequence spaces, $ C $ is always a positive operator. Hence, its order spectrum is well defined within the Banach algebra of all regular operators on $ X .$ The purpose of this note is to show, for every $ X $ belonging to the above list of Banach lattice sequence spaces, that the order spectrum $ σ_{\rm o} (C)$ of $ C $ coincides with its usual spectrum $ σ( C)$ when $ C $ is considered as a continuous linear operator on the Banach space $ X .$

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arXiv2018

The Cesàro operator on duals of power series spaces of infinite type

A detailed investigation is made of the continuity, spectrum and mean ergodic properties of the Cesàro operator $C$ when acting on the strong duals of power series spaces of infinite type. There is a dramatic difference in the nature of the spectrum of $C$ depending on whether or not the strong dual space (which is always Schwartz) is nuclear.

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arXiv2018

Schauder bases and the decay rate of the heat equation

We consider the classical Cauchy problem for the linear heat equation and integrable initial data in the Euclidean space $\mathbb{R}^N$. In the case $N=1$ we show that given a weighted $L^p$-space $L_w^p(\mathbb{R})$ with $1 \leq p < \infty$ and a fast growing weight $w$, there is a Schauder basis $(e_n)_{n=1}^\infty$ in $L_ w^p(\mathbb{R})$ with the following property: given a positive integer $m $ there exists $n_m > 0$ such that, if the initial data $f$ belongs to the closed linear space of $e_n$ with $n \geq n_m$, then the decay rate of the solution of the heat equation is at least $t^{-m}$. The result is also generalized to the case $N >1$ with a slightly weaker formulation. The proof is based on a construction of a Schauder basis of $L_w^p( \mathbb{R}^N)$, which annihilates an infinite sequence of bounded functionals.

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arXiv2017

Monomial basis in Korenblum type spaces of analytic functions

It is shown that the monomials $Λ=(z^n)_{n=0}^{\infty}$ are a Schauder basis of the Fréchet spaces $A_+^{-γ}, \ γ\geq 0,$ that consists of all the analytic functions $f$ on the unit disc such that $(1-|z|)^μ|f(z)|$ is bounded for all $μ> γ$. Lusky \cite{L} proved that $Λ$ is not a Schauder basis for the closure of the polynomials in weighted Banach spaces of analytic functions of type $H^{\infty}$. A sequence space representation of the Fréchet space $A_+^{-γ}$ is presented. The case of (LB)-spaces $A_{-}^{-γ}, \ γ> 0,$ that are defined as unions of weighted Banach spaces is also studied.

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arXiv2017

Solid hulls and cores of weighted $H^\infty$-spaces

We determine the solid hull and solid core of weighted Banach spaces $H_v^\infty$ of analytic functions $f$ such that $v|f|$ is bounded, both in the case of the holomorphic functions on the disc and on the whole complex plane, for a very general class of radial weights $v$. Precise results are presented for concrete weights on the disc that could not be treated before. It is also shown that if $H_v^\infty$ is solid, then the monomials are an (unconditional) basis of the closure of the polynomials in $H_v^\infty$. As a consequence $H_v^\infty$ does not coincide with its solid hull and core in the case of the disc. An example shows that this does not hold for weighted spaces of entire functions.

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arXiv2017

The Cesàro operator in weighted $\ell_1$ spaces

Unlike for $\ell_p$, $1<p\leq\infty$, the discrete Cesàro operator $C$ does not map $\ell_1$ into itself. We identify precisely those weights $w$ such that $C$ does map $\ell_1(w)$ continuously into itself. For these weights a complete description of the eigenvalues and the spectrum of $C$ are presented. It is also possible to identify all $w$ such that $C$ is a compact operator in $\ell_1(w)$. The final section investigates the mean ergodic properties of $C$ in $\ell_1(w)$. Many examples are presented in order to supplement the results and to illustrate the phenomena that occur.

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arXiv2017

A note on completeness of weighted normed spaces of analytic functions

Given a non-negative weight $v$, not necessarily bounded or strictly positive, defined on a domain $G$ in the complex plane, we consider the weighted space $H_v^\infty(G)$ of all holomorphic functions on $G$ such that the product $v|f|$ is bounded in $G$ and study the question of when is such a space complete under the canonical sup-seminorm. We obtain both some necessary and some sufficient conditions in terms of the weight $v$, exhibit several relevant examples, and characterize completeness in the case of spaces with radial weights on balanced domains.

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arXiv2017

The Cesàro operator on power series spaces

The discrete Cesàro operator $\mathsf{C}$ is investigated in the class of power series spaces $Λ_0(α)$ of finite type. Of main interest is its spectrum, which is distinctly different when the underlying Fréchet space $Λ_0(α)$ is nuclear as for the case when it is not. Actually, the nuclearity of $Λ_0(α)$ is characterized via certain properties of the spectrum of $\mathsf{C}$. Moreover, $\mathsf{C}$ is always power bounded, uniformly mean ergodic and, whenever $Λ_0(α)$ is nuclear, also has the property that the range $(I-\mathsf{C})^m(Λ_0(α))$ is closed in $Λ_0(α)$, for each $m\in\mathbb{N}$.

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arXiv2016

The Cesaro operator in growth Banach spaces of analytic functions

The Cesaro operator $\mathsf{C}$, when acting in the classical growth Banach spaces $A^{-γ}$ and $A_0^{-γ}$, for $γ> 0 $, of analytic functions on $\mathbb{D}$, is investigated. Based on a detailed knowledge of their spectra (due to A. Aleman and A.-M. Persson) we are able to determine the norms of these operators precisely. It is then possible to characterize the mean ergodic and related properties of $\mathsf{C}$ acting in these spaces. In addition, we determine the largest Banach space of analytic functions on $\mathbb{D}$ which $\mathsf{C}$ maps into $A^{-γ}$ (resp. into $A_0^{-γ}$); this optimal domain space always contains $A^{-γ}$ (resp. $A_0^{-γ}$) as a proper subspace.

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arXiv2016

Solid hulls of weighted Banach spaces of entire functions

Given a continuous, radial, rapidly decreasing weight $v$ on the complex plane $\mathbf{C}$, we study the solid hull of its associated weighted space $H_v^\infty(\mathbf{C})$ of all the entire functions $f$ such that $v|f|$ is bounded. The solid hull is found for a large class of weights satisfying the condition (B) of Lusky. Precise formulations are obtained for weights of the form $v(r)=\exp(-ar^p), a>0, p>0$. Applications to spaces of multipliers are included.

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arXiv2015

Abscissas of weak convergence of vector valued Dirichlet series

The abscissas of convergence, uniform convergence and absolute convergence of vector valued Dirichlet series with respect to the original topology and with respect to the weak topology $σ(X,X')$ of a locally convex space $X$, in particular of a Banach space $X$, are compared. The relation of their coincidence with geometric or topological properties of the underlying space $X$ is investigated. Cotype in the context of Banach spaces, and nuclearity and certain topological invariants for Fréchet spaces play a relevant role.

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arXiv2014

A note about Volterra operators on weighted Banach spaces of entire functions

We characterize boundedness, compactness and weak compactness of Volterra operators acting between different weighted Banach spaces of entire functions with weighted sup-norms in terms of the symbol g. Thus we complement recent work by Bassallote, Contreras, Hernández-Mancera, Martín and Paul for spaces of holomorphic functions on the disc and by Constantin and Peláez for reflexive weighted Fock spaces.

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arXiv2014

Frames and representing systems in Fréchet spaces and their duals

Frames and Bessel sequences in Fréchet spaces and their duals are defined and studied. Their relation with Schauder frames and representing systems is analyzed. The abstract results presented here, when applied to concrete spaces of analytic functions, give many examples and consequences about sampling sets and Dirichlet series expansions.

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