On recurrent sets of operators

TL;DR

This paper generalizes the concept of recurrence from individual operators to sets of operators on Banach spaces, and explores the recurrence properties of $C$-regularized groups of operators.

Contribution

It introduces the notion of recurrence for sets of operators and applies it to analyze $C$-regularized groups, extending existing theory.

Findings

Generalization of recurrence to sets of operators

Application to $C$-regularized groups

New insights into operator recurrence behavior

Abstract

An operator $T$ acting on a Banach space $X$ is said to be recurrent if for each $U$; a nonempty open subset of $X$, there exists $n\in\mathbb{N}$ such that $T^n(U)\cap U\neq\emptyset.$ In the present work, we generalize this notion from a single operator to a set $\Gamma$ of operators. As application, we study the recurrence of $C$-regularized group of operators.