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Restrictions of bounded linear operators: Closed range

Author: Bruce A. Barnes
Journal: Proc. Amer. Math. Soc. 135 (2007), 1735-1740
MSC (2000): Primary 47A05
Published electronically: November 15, 2006
MathSciNet review: 2286083
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Abstract: Let $ T$ be a bounded linear operator on a Banach space $ X,$ and let $ W$ be a subspace of $ X$ which is a Banach space and $ T-$ invariant. Denote by $ T_{W}$ the restriction of $ T$ to $ W. $ This paper explores the questions:

$ \circ $ If the range of $ T$ is closed, under what conditions is the range of $ T_{W}$ closed?

$ \circ $ If the range of $ T_{W}$ is closed, under what conditions is the range of $ T$ closed?

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Additional Information

Bruce A. Barnes
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403

Keywords: Restriction of a bounded linear operator, closed range.
Received by editor(s): September 9, 2005
Received by editor(s) in revised form: December 28, 2005
Published electronically: November 15, 2006
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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