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Proceedings of the American Mathematical Society

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

 

 

A note on transforms of subspaces of Hilbert space


Author: W. E. Longstaff
Journal: Proc. Amer. Math. Soc. 76 (1979), 268-270
MSC: Primary 47B99; Secondary 46C10
MathSciNet review: 537086
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Abstract: The transform of a family $ \mathcal{F}$ of (closed linear) subspaces of a Hilbert space H by an invertible (bounded, linear) operator S on H is the family of subspaces $ \{ SM:M \in \mathcal{F}\} $. It is shown that the set of projections $ \{ {P_{SM}}:M \in \mathcal{F}\} $ is closed in the uniform (respectively, strong, weak) operator topology if and only if the set of projections $ \{ {P_M}:M \in \mathcal{F}\} $ is uniformly (respectively, strongly, weakly) closed. This answers affirmatively a question raised by K. J. Harrison.


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DOI: https://doi.org/10.1090/S0002-9939-1979-0537086-9
Article copyright: © Copyright 1979 American Mathematical Society