
María Del Pilar Romero de la RosaUniversidad de Cádiz | UCA · Department of Mathematics
María Del Pilar Romero de la Rosa
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Publications (29)
A continuous linear operator L defined on the space of entire functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}({\mathbb {C}})$$\end{document} is s...
An operator T acting on a separable complex Banach space $$\mathcal {B}$$ B is said to be hypercyclic if there exists $$f\in \mathcal {B}$$ f ∈ B such that the orbit $$\{T^n f:\ n\in \mathbb {N}\}$$ { T n f : n ∈ N } is dense in $$\mathcal {B}$$ B . Godefroy and Shapiro (J. Funct. Anal., 98(2):229–269, 1991) characterized those elements, which are...
A continuous linear operator T T defined on a Fréchet space X X is said to be hypercyclic if there exists f ∈ X f\in X such that, the orbit { T n f } \left\{{T}^{n}f\right\} is dense in X X . In this article, we consider the operators introduced by Aron and Markose, defined on the space of entire functions by T λ , b f ( z ) = f ′ ( λ z + b ) {T}_{...
We correct a logic mistake in our paper “On statistical convergence and strong Cesàro convergence by moduli for double sequences” (León-Saavedra et al. in J. Inequal. Appl. 2022:62, 2022).
We correct a logic mistake in our paper “On statistical convergence and strong Cesàro convergence by moduli” (León-Saavedra et al. in J. Inequal. Appl. 23:298, 2019).
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...
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...
We present Korovkin approximation theorems that incorporate summability methods. These result allows us to obtain a unified treatment of several previous results, focusing on the underlying structure and the properties that a summability method should satisfy in order to establish a Korovkin-type approximation result. As a by-product we obtain new...
In earlier works, F. León and coworkers discovered a remarkable structure between statistical convergence and strong Cesàro convergence, modulated by a function f (called a modulus function). Such nice structure pivots around the notion of compatible modulus function. In this paper, we will explore such a structure in the framework of lacunary stat...
Here we fully complete the studies initiated by Vinod K. Bhardwaj and Shweta Dhawan in \cite{hindawi} which relate different convergence methods which involves the classical statistical and the classical strong Ces\`aro convergences by means of lacunary sequences and measures of density in $\mathbb{N}$ modulated by a modulus function $f$.
A continuous linear operator on a Fréchet space X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {X}$$\end{document} is frequently hypercyclic if there exists...
A continuous linear operator L defined on the space of entire functions H(C) is said to be an extended $lambda$-eigenoperator of the differentiation operator D provided DL = $lambda$LD. Here we fully characterize when an extended $lambda$-eigenoperator of D is supercyclic, it has a hypercyclic subspace or it has a supercyclic subspace.
A remarkable result on summability states that the statistical convergence and the strong Cesàro convergence are closely connected. Given a modulus function f , we will establish that a double sequence that is f -strong Cesàro convergent is always f -statistically convergent. The converse, in general, is false even for bounded sequences. However, w...
Пусть $\alpha$ - комплексный скаляр, а $A$ - ограниченный линейный оператор в гильбертовом пространстве $H$. Говорят, что $\alpha$ является расширенным собственным значением оператора $A$, если существует ненулевой ограниченный линейный оператор $X$, такой, что выполняется равенство $AX=\alpha XA$. В весовых пространствах Харди, инвариантных относи...
An operator T acting on a separable F-space X is called hypercyclic if there exists f∈X such that the orbit {Tnf} is dense in X. Here we determine when an operator that λ-commutes with the operator of differentiation on the space of entire functions is hypercyclic, extending results by G. Godefroy and J. H. Shapiro [16] and R. M. Aron and D. Markos...
We survey some results that were originated studying extended eigenvalues of bounded linear operators and their corresponding extended eigenoperators. We cover some aspects of operator theory, such as spectral theory, commutants and bicommutans, orbits of linear operators, etc. We review some recent results and we end up with open questions and fut...
A closed subspace S of ℓ∞ is said to be a ℓ∞-Grothendieck subspace if c0⊂S (hence ℓ∞⊂S⁎⁎) and every σ(S⁎,S)-convergent sequence in S⁎ is σ(S⁎,ℓ∞)-convergent. Here we give examples of closed subspaces of ℓ∞ containing c0 which are or fail to be ℓ∞-Grothendieck.
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 $\sigma(S^*,S)$-convergent sequence in $S^*$ is $\sigma(S^*,\ell_\infty)$-convergent. Here we give examples of closed subspaces of $\ell_\infty$ containing $c_0$ which are or fail to be $\...
In this note, we prove a Schur-type lemma for bounded multiplier series. This result allows us to obtain a unified vision of several previous results, focusing on the underlying structure and the properties that a summability method must satisfy in order to establish a result of Schur’s lemma type.
We characterize when the Cesàro means of higher order for Banach spaces operators are hypercyclic. This is a useful tool to prove that an operator is convex-cyclic and it provides a large number of examples of convex-cyclic operators. A complex number \(\lambda \) is said to be an extended eigenvalue of a bounded linear operator T if there exists a...
Abstract In this paper we will establish a result by Connor, Khan and Orhan (Analysis 8:47–63, 1988; Publ. Math. (Debr.) 76:77–88, 2010) in the framework of the statistical convergence and the strong Cesàro convergence defined by a modulus function f. Namely, for every modulus function f, we will prove that a f-strongly Cesàro convergent sequence i...
We aim to unify several results which characterize when a series is weakly unconditionally Cauchy (wuc) in terms of the completeness of a convergence space associated to the wuc series. If, additionally, the space is completed for each wuc series, then the underlying space is complete. In the process the existing proofs are simplified and some unan...
This paper unifies several versions of the Orlicz–Pettis theorem that incorporate summability methods. We show that a series is unconditionally convergent if and only if the series is weakly subseries convergent with respect to a regular linear summability method. This includes results using matrix summability, statistical convergence with respect...
For an operator T acting on a complex infinite dimensional Banach space X such that T⊕T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\oplus T$$\end{document} is cycl...
A continuous linear operator T on a Frechet space F is hypercyclic if there exists a vector f is an element of F (which is called hypercyclic for T) such that the orbit {T(n)f : n is an element of N} is dense in F. A subset M of a vector space F is spaceable if M boolean OR {0} contains an infinite-dimensional closed vector space. In this paper not...
A bounded operator on a Banach space is convex cyclic if there exists a vector such that the convex hull generated by the orbit is dense in . In this note we study some questions concerned with convex-cyclic operators. We provide an example of a convex-cyclic operator such that the power fails to be convex cyclic.
Using this result we solve three...
An algebra of bounded linear operators on a Banach space is said to be {\em
strongly compact} if its unit ball is precompact in the strong operator
topology, and a bounded linear operator on a Banach space is said to be {\em
strongly compact} if the algebra with identity generated by the operator is
strongly compact. Our interest in this notion ste...
Let A be a bounded linear operator defined on a separable Banach space X. Then A is said to be supercyclic if there exists a vector x ∈ X (later called supercyclic for A), such that the projective orbit $\{\lambda A^{n} x\,:\,n \in {\mathbb{N}},\,\lambda \in
{\mathbb{C}}\}$\{\lambda A^{n} x\,:\,n \in {\mathbb{N}},\,\lambda \in
{\mathbb{C}}\} is den...