Local Ergodic Theorems for $$C_0$$-Semigroups

Abstract

Let \(\{T(t)\}_{t\ge 0}\) 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 non-empty 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.