Operator theoretic properties of semigroups in terms of their generators

S. Blunck¹ ; L. Weis¹

¹ Mathematisches Institut I Universität Karlsruhe Englerstr. 2 D-76128 Karlsruhe, Germany

Studia Mathematica, Tome 146 (2001) no. 1, pp. 35-54

Résumé

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 $D(A) \hookrightarrow X$.

Détail

DOI : 10.4064/sm146-1-3

Keywords: semigroup generator banach space cal operator ideal class compact hilbert schmidt trace class operators resolvent lambda belongs cal only integrated semigroup int belongs cal analytic semigroups cal implies cal precise estimates growth cal norm trace terms resolvent growth imbedding hookrightarrow