Frequently recurrence properties and block families

TL;DR

This paper investigates the spectral properties of reiteratively hypercyclic operators, establishing their perfect spectrum, and explores $ ext{F}$-recurrence and block families to understand operator dynamics in Banach spaces.

Contribution

It proves that reiteratively hypercyclic operators have perfect spectrum and introduces the study of $ ext{F}$-recurrence and block families in this context.

Findings

Reiteratively hypercyclic operators have perfect spectrum.

Existence of Banach spaces without any reiteratively hypercyclic operators.

Analysis of $ ext{F}$-recurrence and block families in operator dynamics.

Abstract

We prove that reiteratively hypercyclic operators have perfect spectrum. Consequently, it follows that there exist separable infinite dimensional Banach spaces that do not support any reiteratively hypercyclic operator. For this, we study $\mathcal F$-recurrence and almost $\mathcal {F}$-recurrence of operators for general families and in particular for a special class of families, called block families.