Abel ergodic theorems for strongly continuous cosine operators

Abstract

Let $(C(t))_{t\in \mathbb {R}}$ be a strongly continuous cosine operators on a Banach space X into itself and let A be their infinitesimal generator. In this work, we study the uniform convergence of the Abel averages $\lambda \int _{0}^{\infty }e^{-\lambda t}C(t)dt$ of C(t) when $\lambda$ tends to $0^+$. More precisely, we show that C(t) is uniformly Abel ergodic if and only if ${X}= {\mathcal {R}}(A)\oplus {\mathcal {N}}(A)$, with ${\mathcal {R}}(A)$ and ${\mathcal {N}}(A)$ the range and the kernel of A, respectively. Next, we examine this theory with the uniform power convergence of the Abel average $\lambda ^2 R(\lambda ^2, A)$, for some $\lambda >0$, where $R(\lambda ^2,A)$ be the resolvent function of A.