Hypercyclicity for the Elements of the Commutant of an Operator

ABSTRACT:Given a bounded linear operator T acting on a complex Banach space, we obtain a spectral condition implying that each operator in the commutant of T different from ?I has a hypercyclic multiple, and we show several examples of operators satisfying this condition. We emphasize that for some of these examples we do not have a description of the commutant of T

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

Extended eigenvalues of composition operators

A complex scalar lambda is said to be an extended eigenvalue of a bounded linear operator A on a complex Hilbert space if there is a nonzero operator X such that AX = lambda XA. The results in this paper provide a full solution to the problem of computing the extended eigenvalues for those composition operators C-phi induced on the Hardy space H-2(D) by linear fractional transformations phi of the unit disk. (c) 2021 Universidad de Sevilla. Published by Elsevier IncThe first and fourth authors were supported by Ministerio de Ciencia, Innovacion y Universidades under Grant MTM 2015-63699-P, and by Fondo Europeo de Desarrollo Regional under Grant PGC2018-094215-B-I00. The second author was also supported by Fondo Europeo de Desarrollo Regional under Grant PGC2018-101514-B-I00, and by Junta de Andalucia under Grant FEDER-UCA18-108415. The first author visited Western Michigan University at an early stage of this research. He would like to acknowledge the warm hospitality received from the third author. He is grateful to the Mathematics Department of WMU for covering local expenses and for the commitment to this global engagement.The third author visited Universidad de Sevilla several times at later stages of this research. He would like to thank the first and fourth authors for making possible this cooperation. He also acknowledges the Ministerio de Ciencia, Innovacion y Universidades for the sponsorship under Grant MTM 2015-6399-P, and the people at Instituto de Matematicas for the organization of his visits

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 T X = 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 Ces`aro operator C0, the finite continuous Ces`aro operator C1 and the infinite continuous Ces`aro operator C∞ defined on the complex Banach spaces ℓ p , Lp [0, 1] and L p [0, ∞) for 1 < p < ∞ by the expressions (C0f)(n): = 1 n + 1 Xn k=0 f(k), (C1f)(x): = 1 x Z x 0 f(t) dt, (C∞f)(x): = 1 x Z x 0 f(t) dt. It is shown that the set of extended eigenvalues for C0 is the interval [1, ∞), for C1 it is the interval (0, 1], and for C∞ it reduces to the singleton {1}.Ministerio de Economía y CompetitividadJunta de AndalucíaVicerrectorado de investigación (Universidad de Cádiz

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 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

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)