Operator theoretic properties of semigroups in terms of their generators
Volume 146 / 2001
Studia Mathematica 146 (2001), 35-54
MSC: 47A60, 47B10, 47D06.
DOI: 10.4064/sm146-1-3
Abstract
Let $(T_t)$ be a ${\rm C}_{0}$ semigroup with generator $A$ on a Banach space $X$ and let ${\cal A}$ be an operator ideal, e.g. the class of compact, Hilbert–Schmidt or trace class operators. We show that the resolvent $R(\lambda ,A)$ of $A$ belongs to ${\cal A}$ if and only if the integrated semigroup $S_t:=\int _0^t T_s\, ds$ belongs to $ {\cal A}$. For analytic semigroups, $S_t\in {\cal A}$ implies $T_t\in {\cal A}$, and in this case we give precise estimates for the growth of the ${\cal A}$-norm of $T_t$ (e.g. the trace of $T_{t}$) in terms of the resolvent growth and the imbedding $D(A) \hookrightarrow X$.
