This is an announcement for the paper "Additive maps preserving the reduced minimum modulus of Banach space operators" by Abdellatif Bourhim.
Abstract: 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.
Archive classification: math.FA math.SP
Mathematics Subject Classification: Primary 47B49; Secondary 47B48, 46A05, 47A10
Remarks: The abstract of this paper was posted on May 2009 in the web page of
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http://front.math.ucdavis.edu/0910.0283
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http://arXiv.org/abs/0910.0283
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