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

 

The spectral properties of certain linear operators and their extensions


Author: Bruce A. Barnes
Journal: Proc. Amer. Math. Soc. 128 (2000), 1371-1375
MSC (1991): Primary 47A10, 47A30
Published electronically: August 5, 1999
MathSciNet review: 1664321
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Abstract: Let $H$ be a Hilbert space with inner-product $(x,y)$, and let $R$ be a bounded positive operator on $H$ which determines an inner-product, $\langle x,y\rangle=(Rx,y),\ x, y\in H$. Denote by $H^-$ the completion of $H$ with respect to the norm $\|x\|=\langle x,x\rangle^{1/2}$. In this paper, operators having certain relationships with $R$ are studied. In particular, if $T=SR^{1/2}$ where $S\in B(H)$, then $T$ has an extension $T^-\in B(H^-)$, and $T$ and $T^-$ have essentially the same spectral and Fredholm properties.


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Additional Information

Bruce A. Barnes
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: barnes@math.uoregon.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05326-5
PII: S 0002-9939(99)05326-5
Keywords: Bounded extension, spectrum, symmetrizable
Received by editor(s): June 23, 1998
Published electronically: August 5, 1999
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society