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

 

Linear equations in subspaces of operators


Authors: R. L. Moore and T. T. Trent
Journal: Proc. Amer. Math. Soc. 128 (2000), 781-788
MSC (1991): Primary 47D25
Published electronically: March 3, 1999
MathSciNet review: 1626454
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Abstract: Given a subspace $\mathcal{S}$ of operators on a Hilbert space, and given two operators $X$ and $Y$ (not necessarily in $\mathcal{S}$), when can we be certain that there is an operator $A$ in $\mathcal{S}$ such that $AX=Y$? If there is one, can we find some bound for its norm? These questions are the subject of a number of papers, some by the present authors, and mostly restricted to the case where $\mathcal{S}$ is a reflexive algebra. In this paper, we relate the broader question involving operator subspaces to the question about reflexive algebras, and we examine a new method of forming counterexamples, which simplifies certain constructions and answers an unresolved question. In particular, there is a simple set of conditions that are necessary for the existence of a solution in the reflexive algebra case; we show that -- even in the case where the co-rank of $X$ is one-these conditions are not in general sufficient.


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Additional Information

R. L. Moore
Affiliation: Department of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487-0350

T. T. Trent
Affiliation: Department of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487-0350

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05064-9
PII: S 0002-9939(99)05064-9
Received by editor(s): April 22, 1998
Published electronically: March 3, 1999
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society