MEAN STABILITY OF SEMIGROUPS
Shmuel Kantorovitz, Sergey Igorevich Piskarev · Taiwanese Journal of Mathematics · 2002
Let $T(\cdot)$ be a bounded $C_0$-semigroup on the Banach space $X$, with generator $A$. It is shown that the denseness of range $A$ is necessary and sufficient for the semigroup's mean stability with respect to suitable weights. Analogous results are valid for power bounded operators, tensor product semigroups, and cosine operator functions.