  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.
