ON ERGODIC AVERAGES AND THE RANGE OF A CLOSED OPERATOR

Abstract

For a $\gamma$-th order Cesàro mean bounded linear operator $T$ on a Banach space $X$, we characterize the range $R(A)$ of the operator $A = T-I$, by using an $A$-ergodic net and its companion net which were introduced by Dotson and developed by Shaw. Similarly, if $A$ is the generator of a $\gamma$-th order Cesàro mean bounded $C_{0}$-semigroup (or strongly continuous cosine operator function) of bounded linear operators on $X$, then we characterize the range $R(A)$.