Local Ergodic Theorems for C0-Semigroups

Abstract

Let {T (t)} t≥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 ∈ X and we establish some conditions implying that T (t) is a local mean ergodic at x.