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Operator theoretic properties of semigroups in terms of their generators

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Blunck S., Weis L.

Studia Mathematica

2001

tom 146

nr 1

35-54

Let $(T_{t})$ be a C₀ semigroup with generator A on a Banach space X and let 𝓐 be an operator ideal, e.g. the class of compact, Hilbert-Schmidt or trace class operators. We show that the resolvent R(λ,A) of A belongs to 𝓐 if and only if the integrated semigroup $S_{t}: = ∫_{0}^{t} T_{s}ds$ belongs to 𝓐. For analytic semigroups, $S_{t} ∈ 𝓐$ implies $T_{t} ∈ 𝓐$, and in this case we give precise estimates for the growth of the 𝓐-norm of $T_{t}$ (e.g. the trace of $T_{t}$) in terms of the resolvent growth and the imbedding D(A) ↪ X.

Ograniczanie wyników

1 Studia Mathematica

1 Blunck S.

1 Weis L.

1 2001

Wyniki wyszukiwania

Operator theoretic properties of semigroups in terms of their generators