Local Ergodic Theorems for C0-Semigroups

TL;DR

This paper investigates the properties of $C_0$-semigroups, showing that uniform ergodicity implies certain spectral properties of their generators and introducing local mean ergodicity at vectors.

Contribution

It establishes a link between uniform ergodicity of $C_0$-semigroups and spectral properties of their generators, and introduces the concept of local mean ergodicity.

Findings

Uniform ergodicity implies the generator lacks the single valued extension property.

The generator must have a nonempty interior of the point spectrum.

Conditions for local mean ergodicity at a vector are established.

Abstract

Let $\{T(t)\}_{t\geq 0}$ be a $C_0$-semigroup of bounded linear operators on the Banach space ${X}$ into itself and let $A$ be their infinitesimal generator. In this paper, we show that if $T(t)$ is uniformly ergodic, then $A$ does not have the single valued extension property, which implies that $A$ must have a nonempty interior of the point spectrum. Furthermore, we introduce the local mean ergodic for $C_0$-semigroup $T(t)$ at a vector $x\in X$ and we establish some conditions implying that $T(t)$ is a local mean ergodic at $x$.