ACTA issues

## Additive maps preserving the reduced minimum modulus of operators

Mohamed Bendaoud

Acta Sci. Math. (Szeged) 77:1-2(2011), 337-346
96/2009

 Abstract. Let $X$ be an infinite dimensional complex Banach space, and let ${\cal L}(X)$ be the algebra of all bounded linear operators on $X$. In this paper, we prove that an additive surjective map $\phi\colon {\cal L}(X)\rightarrow{\cal L}(X)$ preserves the reduced minimum modulus if and only if either there exist bijective isometries $U\colon X\rightarrow X$ and $V\colon X\rightarrow X$ both linear or both conjugate linear such that $\phi(T)=UTV$ for all $T\in{\cal L}(X)$, or there exist bijective isometries $U\colon X^*\rightarrow X$ and $V\colon X\rightarrow X^*$ both linear or both conjugate linear such that $\phi(T)=UT^*V$ for all $T\in{\cal L}(X)$. AMS Subject Classification (1991): 47B48, 47A10, 46H05 Keyword(s): reduced minimum modulus, isometry, additive preservers Received August 25, 2009, and in final form April 21, 2010. (Registered under 96/2009.)