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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the compactness of strongly continuous semigroups and cosine functions of operators
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by Hernán R. Henríquez PDF
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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 1417-1424
  • MSC: Primary 47D03; Secondary 47B07, 47D09
  • DOI: https://doi.org/10.1090/S0002-9939-1995-1227517-2
  • MathSciNet review: 1227517