Additive maps preserving the reduced minimum modulus of Banach space operators

Details Additive maps preserving the reduced minimum modulus of Banach space operators Additive maps preserving the reduced minimum modulus of Banach space operators

2009-10-01

arxiv.org/abs/0910.0283v1

Mathematics

Functional Analysis

The abstract of this paper was posted on May 2009 in the web page of the analysis group of Laval University (http://newton.m

Scientific paper

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 $\phi$ 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 $\phi(T)=UTV$ 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 $\phi(T)=UT^*V$ 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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