Abstract. Let $X$ and $Y$ be two infinite dimensional real or complex Banach spaces, and $\phi :{\cal L}(X)\to{\cal L}(Y)$ be an additive surjective mapping. We show that if $\phi $ preserves the minimum modulus or the surjectivity modulus, then either there exist two surjective linear or conjugate linear isometries $U$ and $V$ such that $\phi(T)=UTV$ for all $T$, or there exist two surjective linear or conjugate linear isometries $U'$ and $V'$ such that $\phi(T)=U'T^*V'$ for all $T$.

AMS Subject Classification (1991): 47B48, 47A10, 46H05

Keyword(s): additive preservers, minimum modulus, surjectivity modulus

Received March 11, 2009, and in revised form July 2, 2009. (Registered under 44/2009.)