On the Compactness of Strongly Continuous Semigroups and Cosine Functions of Operators
نویسندگان
چکیده
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(-)x : x £ X, \\x\\ < 1} is relatively compact in any space LP([Q, a]) ; X) for 1 < p < oo and a > 0. We establish similar results for (T(t) I)n , n 6 N , and for cosine and sine functions of operators.
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