On Modulated Lacunary Statistical Convergence of Double Sequences

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 statistical convergence for double sequences and discover that such structure remains true for lacunary compatible modulus functions. Thus, we continue the work of Hacer ¸Senül, Mikail Et and Yavuz Altin, and we fully solve some questions posed by them

Growth of hypercyclic entire functions for some non-convolution operators

A continuous linear operator T defined on a Frechet space X is said to be hypercyclic if there exists f ∈ χsuch that, the orbit {Tnf} is dense in X. In this article, we consider the operators introduced by Aron and Markose, defined on the space of entire functions by Tλb f(z) = f(λz+b) λ ∈ ℂ\{1}, b ∈ ℂ, and we aimed to explore the rate of growth of hypercyclic vectors for Tλb. We discover that Tλb, is a weighted backward shift with respect to some basis and this fact allows us to find sharp estimates of the growth of Tλb -hypercyclic vectors. When λ > 1, the Tλb -hypercyclic function growth is similar to the D-hypercyclic functions (D is the differentiation operator), and when λ > 1, theλb -hypercyclic functions can grow very slowly but not arbitrarily slowly. A lower bound of this growth is found in terms of the W-Lambert function. Finally, partial results are obtained for Tλb -frequently hypercyclic functions

Modulated Lacunary Statistical and Strong-Cesàro Convergences

Here, we continued the studies initiated by Vinod K. Bhardwaj and Shweta Dhawan which relate different convergence methods involving the classical statistical and the classical strong Cesàro convergences by means of lacunary sequences and measures of density in (Formula presented.) modulated by a modulus function f. A method for constructing non-compatible modulus functions was also included, which is related to symmetries with respect to (Formula presented.)

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

Hypercyclicity of operators that λ-commute with the hardy backward shift

An operator T acting on a separable complex Banach space B is said to be hypercyclic if there exists f ∈ B such that the orbit {Tnf ∶ n ∈ ℕ} is dense in B. Godefroy and Shapiro (J. Funct. Anal., 98(2):229–269, 1991) characterized those elements, which are hypercyclic, in the commutant of the Hardy backward shift. In this paper, we study some dynamical properties of operators X that 휆-commute with the Hardy backward shift B, that is, BX = 휆XB.17 página

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