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Invertibility preserving linear maps and algebraic reflexivity of elementary operators of length one

Author: Peter Semrl
Journal: Proc. Amer. Math. Soc. 130 (2002), 769-772
MSC (2000): Primary 47B49
Published electronically: August 29, 2001
MathSciNet review: 1866032
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Let $X$ and $Y$ be real or complex Banach spaces. We show that a surjective linear map $\phi:{\mathcal B}(X) \to {\mathcal B}(Y)$ preserving invertibility in both directions is either of the form $\phi(T)=ATB$ or the form $\phi(T)=CT'D$, where $A:X\to Y$, $B:Y\to X$, $C:X' \to Y$, and $D:Y\to X'$ are bounded invertible linear operators. As an application we improve a result of Larson and Sourour on algebraic reflexivity of elementary operators of length one.

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Additional Information

Peter Semrl
Affiliation: Institute of Mathematics, Physics, and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia

Received by editor(s): July 22, 1998
Received by editor(s) in revised form: September 14, 2000
Published electronically: August 29, 2001
Additional Notes: This research was supported in part by a grant from the Ministry of Science of Slovenia
Communicated by: David R. Larson
Article copyright: © Copyright 2001 American Mathematical Society