Proceedings of the American Mathematical Society

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



Minimal rank and reflexivity of operator spaces

Authors: Roy Meshulam and Peter Semrl
Journal: Proc. Amer. Math. Soc. 135 (2007), 1839-1842
MSC (2000): Primary 47L05; Secondary 15A03
Published electronically: December 29, 2006
MathSciNet review: 2286094
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Abstract: Let $ {\mathcal{S}}$ be an $ n$-dimensional space of linear operators between the linear spaces $ U$ and $ V$ over an algebraically closed field $ \mathbb{F}$. Improving results of Larson, Ding, and Li and Pan we show the following.

Theorem. Let $ S_1, \ldots , S_n$ be a basis of $ \mathcal{S}$. Assume that every nonzero operator in $ \mathcal{S}$ has rank larger than $ n$. Then a linear operator $ T : U \to V$ belongs to $ \mathcal{S}$ if and only if for every $ u\in U$, $ Tu$ is a linear combination of $ S_1 u , \ldots , S_n u$.

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Additional Information

Roy Meshulam
Affiliation: Department of Mathematics, Technion, Haifa 32000, Israel

Peter Semrl
Affiliation: Department of Mathematics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia

Received by editor(s): April 7, 2005
Received by editor(s) in revised form: February 10, 2006
Published electronically: December 29, 2006
Additional Notes: The research of the first author was supported in part by the Israel Science Foundation
The research of the second author was supported in part by a grant from the Ministry of Science of Slovenia
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.