Recurrent strongly continuous operator families on Banach space

TL;DR

This paper studies recurrent and rigid $C_0$-semigroups on Banach spaces, providing structural characterizations and conditions for chaos, especially in translation semigroups on weighted function spaces.

Contribution

It introduces the concept of (uniformly) $C_0$-rigid semigroups and characterizes recurrent $C_0$-semigroups induced by semiflows.

Findings

Characterization of Li-Yorke chaos in translation semigroups

Structural description of $C_0$-rigid semigroups

Conditions for recurrence in semigroups induced by semiflows

Abstract

In this paper, we analyze recurrent $C_{0}$-semigroups of bounded operators on Banach spaces. We also introduce the notion of a (uniformly) $C_{0}$-rigid semigroups of bounded operators and give a structural characterization of them. A characterization of a Li-Yorke chaoticity of the translation semigroup $(T(t))_{t\geq 0}$ on weighted spaces of integrable functions and continuous functions in terms of admissible weight function is given. The recurrent $C_0$-semigroups induced by semiflows are characterized on the spaces of integrable functions and of spaces of continuous functions.