On frequently supercyclic operators and an F_{\Gamma}-hypercyclicity criterior with applications

TL;DR

This paper introduces F_{ ext{Gamma}}-hypercyclic operators, providing a criterion for their identification and demonstrating their existence in various operators and spaces, advancing the understanding of hypercyclicity.

Contribution

It defines F_{ ext{Gamma}}-hypercyclic operators, establishes a criterion for their detection, and shows their presence in unilateral pseudo-shift operators and infinite-dimensional Banach spaces.

Findings

Unilateral pseudo-shift operators are F_{ ext{Gamma}}-hypercyclic for unbounded Gamma.

Every separable infinite-dimensional Banach space supports an F_{ ext{Gamma}}-hypercyclic operator.

Characterization of subsets Gamma as hypercyclic scalar sets.

Abstract

Given a Furstenberg family F and a subset {\Gamma} of C, we introduce and explore the notions of F_{\Gamma}-hypercyclic operator and F-hypercyclic scalar set. First, the study of F_C-hypercyclic operators yields new interesting information about frequently supercyclic, U-frequently supercyclic, reiteratively supercyclic and supercyclic operators. Then we provide a criterion for identifying F_{\Gamma}-hypercyclic operators. As applications of this criterion, we show that any unilateral pseudo-shift operator on c_0(N) or l_p(N) is F_{\Gamma}-hypercyclic for every unbounded subset {\Gamma} of C. Moreover, under the same condition on {\Gamma}, we show that any separable infinite-dimensional Banach space supports an F_{\Gamma}-hypercyclic operator. Finally, our study provides sufficient and necessary conditions for a subset {\Gamma} of C to be a hypercyclic scalar set. These results give partial answers to a question raised by Charpentier, Ernst, and Menet in 2016.