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José Bonet

Publications and source records attributed to José Bonet.

At least 19 recordsLinked to original sources
arXiv2026

Optimal domain of Volterra operators in classes of Banach spaces of analytic functions

A thorough investigation is made of the optimal domain space of generalized Volterra operators, Cesàro operators and other operators when they act in various Banach spaces of analytic functions. Of particular interest is the situation when the operators act in Hardy spaces, Korenblum growth spaces and more general weighted spaces. The optimal domain space may be genuinely larger than the initial domain of the operator, or not. In the former case, the initial space may or may not be dense in the optimal domain space. Sometimes the optimal domain space can be identified with a known Banach space of analytic functions, on other occasions it determines a new space.

math.FA↗
arXiv2026

Optimal domain of Volterra operators in Korenblum spaces

The aim of this article is to study the largest domain space $[T,X]$, whenever it exists, of a given continuous linear operator $T\colon X\to X$, where $X\subseteq H(\mathbb{D})$ is a Banach space of analytic functions on the open unit disc $\mathbb{D}\subseteq \mathbb{C}$. That is, $[T,X]\subseteq H(\mathbb{D})$ is the \textit{largest} Banach space of analytic functions containing $X$ to which $T$ has a continuous, linear, $X$-valued extension $T\colon [T,X]\to X$. The class of operators considered consists of generalized Volterra operators $T$ acting in the Korenblum growth Banach spaces $X:=A^{-γ}$, for $γ>0$. Previous studies dealt with the classical Cesàro operator $T:=C$ acting in the Hardy spaces $H^p$, $1\leq p<\infty$, \cite{CR}, \cite{CR1}, in $A^{-γ}$, \cite{ABR-R}, and more recently, generalized Volterra operators $T$ acting in $X:=H^p$, \cite{BDNS}.

math.FA↗
arXiv2024

Mean ergodic and related properties of generalized Cesàro operators in BK-sequence spaces

Recent results concerning the linear dynamics and mean ergodicity of compact operators in Banach spaces, together with additional new results, are employed to investigate various spectral properties of generalized Cesàro operators acting in large classes of classical BK-sequence spaces. Of particular interest is to determine the eigenvalues and the corresponding eigenvectors of such operators and to decide whether (or not) the operators are power bounded, mean ergodic and supercyclic.

math.FA↗
arXiv2024

Cesàro operators associated with Borel measures acting on weighted spaces of holomorphic functions with sup-norm

Let $μ$ be a positive finite Borel measure on $[0,1).$ Cesàro-type operators $C_μ$ when acting on weighted spaces of holomorphic functions are investigated. In the case of bounded holomorphic functions on the unit disc we prove that $C_μ$ is continuous if and only if it is compact. In the case of weighted Banach spaces of holomorphic function defined by general weights, we give sufficient and necessary conditions for the continuity and compactness. For standard weights, we characterize the continuity and compactness on classical growth Banach spaces of holomorphic functions. We also study the point spectrum and the spectrum of $C_μ$ on the space of holomorphic functions on the disc, on the space of bounded holomorphic functions on the disc, and on the classical growth Banach spaces of holomorphic functions. All characterizations are given in terms of the sequence of moments $(μ_n)_{n\in\N_0}$. The continuity, compactness and spectrum of $C_μ$ acting on Fréchet and (LB) Korenblum type spaces are also considered.

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arXiv2024

Generalized Cesàro operators in the disc algebra and in Hardy spaces

Generalized Cesàro operators $C_t$, for $t\in [0,1)$, are investigated when they act on the disc algebra $A(\mathbb{D})$ and on the Hardy spaces $H^p$, for $1\leq p \leq \infty$. We study the continuity, compactness, spectrum and point spectrum of $C_t$ as well as their linear dynamics and mean ergodicity on these spaces.

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arXiv2024

Cesàro operators on the space of analytic functions with logarithmic growth

Continuity, compactness, the spectrum and ergodic properties of Cesàro operators are investigated when they act on the space $VH(\mathbb{D})$ of analytic functions with logarithmic growth on the open unit disc $\mathbb{D}$ of the complex plane. The space $VH(\mathbb{D})$ is a countable inductive limit of weighted Banach spaces of analytic functions with compact linking maps. It was introduced and studied by Taskinen and also by Jasiczak.

math.FA↗
arXiv2024

Generalized Hilbert operators acting on weighted spaces of holomorphic functions with sup-norms

The behaviour of the generalized Hilbert operator associated with a positive finite Borel measure $μ$ on $[0,1)$ is investigated when it acts on weighted Banach spaces of holomorphic functions on the unit disc defined by sup-norms and on Korenblum type growth Banach spaces. It is studied when the operator is well defined, bounded and compact. To this aim, we study when it can be represented as an integral operator. We observe important differences with the behaviour of the Cesàro-type operator acting on these spaces, getting that boundedness and compactness are equivalent concepts for some standard weights. For the space of bounded holomorphic functions on the disc and for the Wiener algebra, we get also this equivalence, which is characterized in turn by the summability of the moments of the measure $μ.$ In the latter case, it is also equivalent to nuclearity. Nuclearity of the generalized Hilbert operator acting on related spaces, such as the classical Hardy space, is also analyzed.

math.FA↗
arXiv2024

Generalized Cesàro operators in weighted Banach spaces of analytic functions with sup-norms

An investigation is made of the generalized Cesàro operators $C_t$, for $t\in [0,1]$, when they act on the space $H(\mathbb{D})$ of holomorphic functions on the open unit disc $\mathbb{D}$, on the Banach space $H^\infty$ of bounded analytic functions and on the weighted Banach spaces $H_v^\infty$ and $H_v^0$ with their sup-norms. Of particular interest are the continuity, compactness, spectrum and point spectrum of $C_t$ as well as their linear dynamics and mean ergodicity.

math.FA↗
arXiv2023

Spectral properties of generalized Cesàro operators in sequence spaces

The generalized Cesàro operators $C_t$, for $t\in [0,1]$, were first investigated in the 1980's. They act continuously in many classical Banach sequence spaces contained in $\mathbb{C}^{\mathbb{N}_0}$, such as $\ell^p$, $c_0$, $c$, $bv_0$, $bv$ and, as recently shown, \cite{CR4}, also in the discrete Cesàro spaces $ces(p)$ and their (isomorphic) dual spaces $d_p$. In most cases $C_t$ ($t\not=1$) is compact and its spectra and point spectrum, together with the corresponding eigenspaces, are known. We study these properties of $C_t$, as well as their linear dynamics and mean ergodicity, when they act in certain non-normable sequence spaces contained in $\mathbb{C}^{\mathbb{N}_0}$. Besides $\mathbb{C}^{\mathbb{N}_0}$ itself, the Fréchet spaces considered are $\ell(p+)$, $ces(p+)$ and $d(p+)$, for $1\leq p<\infty$, as well as the (LB)-spaces $\ell(p-)$, $ces(p-)$ and $d(p-)$, for $1<p\leq\infty$.

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arXiv2022

On solid cores and hulls of weighted Bergman spaces $A_μ^1$

We consider weighted Bergman spaces $A_μ^1$ on the unit disc as well as the corresponding spaces of entire functions, defined using non-atomic Borel measures with radial symmetry. By extending the techniques from the case of reflexive Bergman spaces we characterize the solid core of $A_μ^1$. Also, as a consequence of a characterization of solid $A_μ^1$-spaces we show that, in the case of entire functions, there indeed exist solid $A_μ^1$-spaces. The second part of the paper is restricted to the case of the unit disc and it contains a characterization of the solid hull of $A_μ^1$, when $μ$ equals the weighted Lebesgue measure with weight $v$. The results are based on a duality relation of weighted $A^1$- and $H^\infty$-spaces, the validity of which requires the assumption that $- \log v$ belongs to the class $\mathcal{W}_0$, studied in a number of publications; moreover, $v$ has to satisfy condition $(b)$, introduced by the authors. The exponentially decreasing weight $v(z) = \exp( -1 /(1-|z|)$ provides an example satisfying both assumptions.

math.FA↗
arXiv2022

ConTrip: Consensus Sentiment review Analysis and Platform ratings in a single score

People unequivocally employ reviews to decide on purchasing an item or an experience on the internet. In that regard, the growing significance and number of opinions have led to the development of methods to assess their sentiment content automatically. However, it is not straightforward for the models to create a consensus value that embodies the agreement of the different reviews and differentiates across equal ratings for an item. Based on the approach proposed by Nguyen et al. in 2020, we derive a novel consensus value named ConTrip that merges their consensus score and the overall rating of a platform for an item. ConTrip lies in the rating range values, which makes it more interpretable while maintaining the ability to differentiate across equally rated experiences. ConTrip is implemented and freely available under MIT license at https://github.com/pepebonet/contripscore

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arXiv2021

Dynamics of the Volterra-type integral and differentiation operators on generalized Fock spaces

Various dynamical properties of the differentiation and Volterra-type integral operators on generalized Fock spaces are studied. We show that the differentiation operator is always supercyclic on these spaces. We further characterize when it is hypercyclic, power bounded and uniformly mean ergodic. We prove that the operator satisfies the Ritt's resolvent condition if and only if it is power bounded and uniformly mean ergodic. Some similar results are obtained for the Volterra-type and Hardy integral operators.

math.FA↗
arXiv2020

Fréchet and (LB) sequence spaces induced by dual Banach spaces of discrete Cesàro spaces

The Fréchet (resp.\ (LB)) sequence spaces $ces(p+) := \cap_{r > p} ces(r), 1 \leq p < \infty $ (resp.\ $ ces (p-) := \cup_{ 1 < r < p} ces (r), 1 < p \leq \infty),$ are known to be very different to the classical sequence spaces $ \ell_ {p+} $ (resp., $ \ell_{p_{-}}).$ Both of these classes of non-normable spaces $ ces (p+), ces (p-)$ are defined via the family of reflexive Banach sequence spaces $ ces (p), 1 < p < \infty .$ The dual Banach spaces $ d (q), 1 < q < \infty ,$ of the discrete Cesàro spaces $ ces (p), 1 < p < \infty,$ were studied by G.\ Bennett, A.\ Jagers and others. Our aim is to investigate in detail the corresponding sequence spaces $ d (p+) $ and $ d (p-),$ which have not been considered before. Some of their properties have similarities with those of $ ces (p+), ces (p-)$ but, they also exhibit differences. For instance, $ ces (p+)$ is isomorphic to a power series Fréchet space of order 1, whereas $ d (p+) $ is isomorphic to such a space of infinite order. Every space $ ces (p+), ces (p-) $ admits an absolute basis but, none of the spaces $ d (p+), d (p-)$ have any absolute basis.

math.FA↗
arXiv2020

On the boundedness of Toeplitz operators with radial symbols over weighted sup-norm spaces of holomorphic functions

We prove sufficient conditions for the boundedness and compactness of Toeplitz operators $T_a$ in weighted sup-normed Banach spaces $H_v^\infty$ of holomorphic functions defined on the open unit disc $\mathbb{D}$ of the complex plane; both the weights $v$ and symbols $a$ are assumed to be radial functions on $\mathbb{D}$. In an earlier work by the authors it was shown that there exists a bounded, harmonic (thus non-radial) symbol $a$ such that $T_a$ is not bounded in any space $H_v^\infty$ with an admissible weight $v$. Here, we show that a mild additional assumption on the logarithmic decay rate of a radial symbol $a$ at the boundary of $\mathbb{D} $ guarantees the boundedness of $T_a$. The sufficient conditions for the boundedness and compactness of $T_a$, in a number of variations, are derived from the general, abstract necessary and sufficient condition recently found by the authors. The results apply for a large class of weights satisfying the so called condition$(B)$, which includes in addition to standard weight classes also many rapidly decreasing weights.

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arXiv2020

Seminar about the Bounded Approximation Property in Fréchet Spaces

The purpose of this seminar, which was presented at the Universitat Politècnica de València in late 2012, is to explain several results concerning the bounded approximation property for Fréchet spaces. We give a full detailed proof of an important result due to Pełczyński that asserts that every separable Fréchet space with the bounded approximation property is isomorphic to a complemented subspace of a Fréchet space with a Schauder basis. We also explain Vogt's example of a nuclear Fréchet space without the bounded approximation property. This example is simpler than the original counterexample due to Dubinski. These examples solved a long standing problem of Grothendieck. Vogt obtained later another simple example of a nuclear Fréchet function space without the bounded approximation property. The relation of the bounded approximation property for Fréchet spaces with a continuous norm and the countably normable spaces, including several results due to Dubinski and Vogt, is also explained.

math.FA↗
arXiv2020

Dynamics of shift operators on non-metrizable sequence spaces

We investigate dynamical properties such as topological transitivity, (sequential) hypercyclicity, and chaos for backward shift operators associated to a Schauder basis on LF-spaces. As an application, we characterize these dynamical properties for weighted generalized backward shifts on Köthe coechelon sequence spaces $k_p((v^{(m)})_{m\in\mathbb{N}})$ in terms of the defining sequence of weights $(v^{(m)})_{m\in\mathbb{N}}$. We further discuss several examples and show that the annihilation operator from quantum mechanics is mixing, sequentially hypercyclic, chaotic, and topologically ergodic on $\mathscr{S}'(\mathbb{R})$.

math.FA↗
arXiv2020

The differentiation operator in the space of uniformly convergent Dirichlet series

Continuity, compactness, the spectrum and ergodic properties of the differentiation operator are investigated, when it acts in the Fréchet space of all Dirichlet series that are uniformly convergent in all half-planes $\{s \in \mathbb{C} \ | \ {\rm Re} s > \varepsilon \}$ for each $\varepsilon>0$. The properties of the formal inverse of the differentiation are also investigated.

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