A note on frequent hypercyclicity of operators that λ -commute with the differentiation operator

A continuous linear operator on a Fréchet space X is frequently hypercyclic if there exists a vector x such that for any nonempty open subset U⊂ X the set of n∈ N∪ { 0 } for which Tnx∈ U has a positive lower density. Here we determine when an operator that commutes up to a factor with the differentiation operator D, defined on the space of entire functions, is frequently hypercyclic

A local spectral condition for strong compactness with some applications to bilateral weighted shifts

An algebra of bounded linear operators on a Banach space is said to be 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 strongly compact if the algebra with identity generated by the operator is strongly compact. Our interest in this notion stems from the work of Lomonosov on the existence of invariant subspaces. We provide a local spectral condition that is sufficient for a bounded linear operator on a Banach space to be strongly compact. This condition is then applied to describe a large class of strongly compact, injective bilateral weighted shifts on Hilbert spaces, extending earlier work of Fernández-Valles and the first author. Further applications are also derived, for instance, a strongly compact, invertible bilateral weighted shift is constructed in such a way that its inverse fails to be a strongly compact operator.Ministerio de Ciencia e InnovaciónJunta de Andalucí

Schur Lemma and Uniform Convergence of Series through Convergence Methods

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

Orlicz–Pettis Theorem through Summability Methods

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 to an ideal, and other variations of summability methods

On statistical convergence and strong Cesaro convergence by moduli for double sequences

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, we will characterize analytically the modulus functions f for which the converse of this result remains true. The results of this paper adapt to several variables the results obtained in (León-Saavedra et al. in J. Inequal. Appl. 12:298, 2019)

On ℓ∞- Grothendieck subspaces

A closed subspace S of l - is said to be a l -Grothendieck subspace if co c S (hence l - c S**) and every [sigma] (S*, S)-convergent sequence in S* is [sigma](S*, l)-convergent. Here we give examples of closed subspaces of l- containing co which are or fail to be l -Grothendiec.The authors were supported by Ministerio de Ciencia, Innovación y Universidades (Spain), grants PGC2018-101514-B-I00, PID2019-103961GB-C22, and by Vicerrectorado de Investigación de la Universidad de Cádiz. This work was also co-financed by the 2014-2020 ERDF Operational Programme, and by the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia, project reference: FEDER-UCA18-108415

Powers of Convex-Cyclic Operators

A bounded operator T on a Banach space X is convex cyclic if there exists a vector x such that the convex hull generated by the orbit Tnxn≥0 is dense in X. In this note we study some questions concerned with convex-cyclic operators. We provide an example of a convex-cyclic operator T such that the power Tn fails to be convex cyclic. Using this result we solve three questions posed by Rezaei (2013)

Hypercyclicity of operators that λ-commute with the differentiation operator on the space of entire functions

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. Markose [1]. © 2022 The AuthorsThe first author is supported by Aula Universitaria del Estrecho, Plan Propio UCA-Internacional . The remaining authors are supported by Ministerio de Ciencia, Innovación y Universidades (Spain), grants MTM2016-76958 , PGC2018-101514-B-I00 , PID2019-103961GB-C22 , and Vicerrectorado de Investigación de la Universidad de Cádiz . This work has been co-financed by the 2014-2020 ERDF Operational Programme and by the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia. Project reference: FEDER-UCA18-10841

On statistical convergence and strong Cesàro convergence by moduli

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 is always f -statistically convergent and uniformly integrable. The converse of this result is not true even for bounded sequences. We will characterize analytically the modulus functions f for which the converse is true. We will prove that these modulus functions are those for which the statistically convergent sequences are f -statistically convergent, that is, we show that Connor–Khan–Orhan’s result is sharp in this sense