Journal of Operator Theory

Volume 67, Issue 1, Winter 2012 pp. 279-288.

Additive maps preserving the reduced minimum modulus of Banach space operators

Authors: Abdellatif Bourhim
Author institution: Departement de mathematiques et de statistique, Universite Laval, Quebec, Canada;
Current address: Department of Mathematics, Syracuse University, 215 Carnegie Building, Syracuse, New York 13244, U.S.A.

Summary: Let $\mathcal{B}(X)$ be the algebra of all bounded linear operators on an infinite dimensional complex Banach space $X$. We prove that an additive surjective map $\varphi$ on $\mathcal{B}(X)$ preserves the reduced minimum modulus if and only if either there are bijective isometries $U:X\to X$ and $V:X\to X$ both linear or both conjugate linear such that $\varphi(T)=UTV$ for all $T\in\mathcal{B}(X)$, or $X$ is reflexive and there are bijective isometries $U:X^*\to X$ and $V:X\to X^*$ both linear or both conjugate linear such that $\varphi(T)=UT^*V$ for all $T\in\mathcal{B}(X)$. As immediate consequences of the ingredients used in the proof of this result, we get the complete description of surjective additive maps preserving the minimum, the surjectivity and the maximum moduli of Banach space operators.

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