Searcharxiv⌕ Search

arXiv subjects

Fernando León-Saavedra

Publications and source records attributed to Fernando León-Saavedra.

8 recordsLinked to original sources
arXiv2025

Minimal commutant and double commutant property for analytic Toeplitz operators

In this paper we study the minimality of the commutant of an analytic Toeplitz operator $M_φ$, when $M_φ$ is defined on the Hardy space $H^2(\mathbb{D})$ and $φ\in H^\infty(\mathbb{D})$, denotes a bounded analytic function on $\mathbb{D}$. Specifically we show that the commutant of $M_φ$ is minimal if and only if the polynomials on $φ$ are weak-star dense in $H^\infty(\mathbb{D})$, that is, $φ$ is a weak-star generator of $H^\infty(\mathbb{D})$. We use our result to characterize when the double commutant of an analytic Toeplitz operator $M_φ$ is minimal, for a large class of symbols $φ$. Namelly, when $φ$ is an entire function, or more generally when $φ$ belongs to the Thomson-Cowen's class.

math.FA↗
arXiv2024

Completeness of a Normed Space via Strong p-Cesàro Summability

In this paper we will characterize the completeness and barrelledness of a normed space through the strong p-Cesàro summability of series. A new characterization of weakly unconditionally Cauchy series and unconditionally convergent series through the strong p-Cesàro summability is obtained.

math.FA↗
arXiv2024

Characterizations of a Banach Space through the Strong Lacunary and the Lacunary Statistical Summabilities

In this manuscript we characterize the completeness of a normed space through the strong lacunary (N-theta) and lacunary statistical convergence (S-theta) of series. A new characterization of weakly unconditionally Cauchy series through N-theta and S-theta is obtained. We also relate the summability spaces associated with these summabilities with the strong p-Cesaro convergence summability space.

math.FA↗
arXiv2023

Classical theorems through convergence methods

We survey some results that provide different versions of classical results through different summability methods. Specifically, in order to adapt such classical results, we analyze which properties should satisfy the summability methods. Sometimes very sharp conditions are obtained, giving a focused view of the subject and from which new problems emerge.

math.FA↗
arXiv2023

Hypercyclicity of operators that $λ$-commute with the Hardy backward shift

An operator $T$ acting on a separable complex Hilbert space $H$ is said to be hypercyclic if there exists $f\in H$ such that the orbit $\{T^n f:\ n\in \mathbb{N}\}$ is dense in $H$. Godefroy and Shapiro \cite{GoSha} characterized those elements in the commutant of the Hardy backward shift which are hypercyclic. In this paper we study some dynamics properties of operators $X$ that $λ$-commute with the Hardy backward shift $B$, that is, $BX=λXB$.

math.FA↗
arXiv2020

On $\ell_\infty$-Grothendieck subspaces

A closed subspace $S$ of $\ell_\infty$ is said to be a \emph{$\ell_\infty$-Grothendieck subspace} if $c_0\subset S$ (hence $\ell_\infty\subset S^{**}$) and every $σ(S^*,S)$-convergent sequence in $S^*$ is $σ(S^*,\ell_\infty)$-convergent. Here we give examples of closed subspaces of $\ell_\infty$ containing $c_0$ which are or fail to be $\ell_\infty$-Grothendieck.

math.FA↗
arXiv2014

Extended eigenvalues for Cesàro operators

A complex scalar $λ$ is said to be an extended eigenvalue of a bounded linear operator $T$ on a complex Banach space if there is a nonzero operator $X$ such that $TX= λXT.$ Such an operator $X$ is called an extended eigenoperator of $T$ corresponding to the extended eigenvalue $λ.$ The purpose of this paper is to give a description of the extended eigenvalues for the discrete Cesàro operator $C_0,$ the finite continuous Cesàro operator $C_1$ and the infinite continuous Cesàro operator $C_\infty$ defined on the complex Banach spaces $\ell^p,$ $L^p[0,1]$ and $L^p[0,\infty)$ for $1 < p <\infty$ by the expressions \begin{align*} (C_0f)(n) \colon & = \frac{1}{n+1} \sum_{k=0}^n f(k),\\ (C_1f)(x) \colon & = \frac{1}{x} \int_0^x f(t)\,dt,\\ (C_\infty f)(x) \colon & = \frac{1}{x} \int_0^x f(t)\,dt. \end{align*} It is shown that the set of extended eigenvalues for $C_0$ is the interval $[1,\infty),$ for $C_1$ it is the interval $(0,1],$ and for $C_\infty$ it reduces to the singleton $\{1\}.$

math.FA↗