Local Ergodic Theorems for C0-Semigroups

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 \(x\).