On Certain forms of Transitivities for Linear Operators

TL;DR

This paper explores various transitivity properties of linear operators, providing new characterizations and criteria, including a generalized Hypercyclicity Criterion and relations between mixing and other transitivity concepts.

Contribution

It introduces a general form of Hypercyclicity Criterion using Furstenberg families and characterizes several transitivity properties for linear operators.

Findings

Characterization of $$-transitive operators via a generalized Hypercyclicity Criterion

Equivalent characterization of mixing operators

Relations between mixing operators and Kitai's Criterion

Abstract

In this article we give several characterizations for various transitivity properties for linear operators. We define a general form of `Hypercyclicity Criterion' using a Furstenberg family $\mathcal{F}$ to characterize $\mathcal{F}$-transitive operators. In particular, we find an equivalent characterization for mixing operators. We study proximal and asymptotic relations for linear operators and prove that the difference between mixing operators and Kitai's Criterion can be presented through these relations. Finally, we find an equivalent characterization of strongly transitive abd strongly product transitive operators.