On the compactness of strongly continuous semigroups and cosine functions of operators

Henríquez, Hernán R.

doi:10.1090/S0002-9939-1995-1227517-2

On the compactness of strongly continuous semigroups and cosine functions of operators
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Proc. Amer. Math. Soc. 123 (1995), 1417-1424 Request permission

Abstract:

In this note we relate two notions of compactness for strongly continuous semigroups of linear operators and cosine functions of linear operators. Specifically, if T denotes a strongly continuous semigroup of linear operators defined on a Banach space X, we will show that T is compact if and only if the set $\{ (T( \bullet )x:x \in X,\left \| x \right \| \leq 1\}$ is relatively compact in any space ${L^p}([0,a]);X)$ for $1 \leq p < \infty$ and $a > 0$. We establish similar results for ${(T(t) - I)^n},n \in {\mathbf {N}}$, and for cosine and sine functions of operators.

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