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

DOI:
https://doi.org/10.1090/S0002-9939-99-05064-9

Published electronically:
March 3, 1999

MathSciNet review:
1626454

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Abstract: Given a subspace of operators on a Hilbert space, and given two operators and (not necessarily in ), when can we be certain that there is an operator in such that ? 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 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 is one-these conditions are not in general sufficient.

**1.**R. G. Douglas,*On Majorization, Factorization, and Range Inclusion of Operators on Hilbert Space*, Proc. Amer. Math. Soc.**17**(1966), 413-415.MR**34:3315****2.**E. Katsoulis, R. Moore, and T. Trent,*Interpolation in Nest Algebras and Applications to Operator Corona Theorems*, Journal of Operator Theory**29**(1993), 115-123. MR**95b:47052****3.**R. Moore and T. Trent,*Solving Operator Equations in Nest Algebras*, Houston Journal of Mathematics, to appear.**4.**R. Moore, and T. Trent,*Interpolation in Inflated Hilbert Spaces*, Proc. Amer. Math. Soc. Journal of Operator Theory, to appear. CMP**98:01**

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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:
https://doi.org/10.1090/S0002-9939-99-05064-9

Received by editor(s):
April 22, 1998

Published electronically:
March 3, 1999

Communicated by:
David R. Larson

Article copyright:
© Copyright 1999
American Mathematical Society