Let 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 B(X) preserves the reduced minimum modulus if and only if either there are bijective isometries U:X -> X and V:X -> X both linear or both conjugate linear such that phi(T)=UTV for all T in B(X), or X is reflexive and there are bijective isometries U:X* -> X and V:X -> X* both linear or both conjugate linear such that phi(T)=UT*V for all T in 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.
Bourhim, Abdellatif, "Additive Maps Preserving the Reduced Minimum Modulus of Banach Space Operators" (2009). Mathematics Faculty Scholarship. Paper 88.
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