Operator theoretic properties of semigroups in terms of their generators

Tom 146 / 2001

S. Blunck, L. Weis Studia Mathematica 146 (2001), 35-54 MSC: 47A60, 47B10, 47D06. DOI: 10.4064/sm146-1-3

Streszczenie

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 $D(A) \hookrightarrow X$.

Autorzy

  • S. BlunckMathematisches Institut I
    Universität Karlsruhe
    Englerstr. 2
    D-76128 Karlsruhe, Germany
    e-mail
  • L. WeisMathematisches Institut I
    Universität Karlsruhe
    Englerstr. 2
    D-76128 Karlsruhe, Germany
    e-mail

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