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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Subspace maps of operators on Hilbert space


Author: W. E. Longstaff
Journal: Proc. Amer. Math. Soc. 84 (1982), 195-201
MSC: Primary 47A05; Secondary 47A10
MathSciNet review: 637168
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Abstract: An operator $ A$ acting on a Hilbert space $ H$ gives rise to a map $ {\varphi _A}$ on the set of subspaces of $ H$ given by $ {\varphi _A}(M) = \overline {AM} $, where $ {\text{'}} - {\text{'}}$ denotes norm closure. This map is called the subspace map of $ A$. By identifying subspaces with projections in the usual way it is shown that for $ A \ne 0$, $ {\varphi _A}$ is uniformly (respectively, strongly) continuous if and only if the approximate point spectrum of $ A$ does not contain 0. In this case it is proved that $ {\varphi _A}$ preserves the property of being uniformly (respectively, strongly, weakly) closed and its effect on reflexivity is described.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0637168-7
PII: S 0002-9939(1982)0637168-7
Article copyright: © Copyright 1982 American Mathematical Society