Continuity of the Restriction of C
0-Semigroups to Invariant Banach Subspaces |
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Authors: | Sander C Hille |
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Institution: | (1) Mathematical Institute, University Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands |
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Abstract: | A linear semigroup in a Banach space induces a linear semigroup on a Banach space that can be continuously embedded in the
former such that its image is invariant. This restriction need not be strongly continuous, although the original semigroup
is strongly continuous. We show that norm or weak compactness of partial orbits is a necessary and sufficient condition for
strong continuity of the restriction of a C0-semigroup. We then show that if the embedded Banach space is reflexive and the norms of the restricted semigroup operators
are bounded near the initial time, then the restricted semigroup is strongly continuous. |
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Keywords: | Primary 47D06 Secondary 46N20 |
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