Cesàro and Abel ergodic theorems for integrated semigroups

Let { )} be an integrated semigroup of bounded linear operators on the Banach space X into itself and let be their generator. In this paper, we consider some necessary and sufficient conditions for the Cesàro mean and the Abel average of ) converge uniformly on B(X). More precisely, we show that the Abel average of ) converges uniformly if and only if X = R( ) ⊕ N( ), if and only if R( ) is closed for some integer and ∥ , ) ∥ → 0 as → 0 , where R( ), N( ) and , ), be the range, the kernel, the resolvent function of , respectively. Furthermore, we prove that if )/ → 0 as → 1, then the Cesàro mean of ) converges uniformly if and only if the Abel average of ) is also converges uniformly.