Nonlinear preservers of group invertible operators

Abstract

Let [Formula: see text] be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space [Formula: see text]. We prove that a bijective bicontinuous map [Formula: see text] on [Formula: see text] preserves the difference of group invertible operators in both directions if and only if [Formula: see text] is either of the form [Formula: see text] or of the form [Formula: see text], where [Formula: see text] is a nonzero scalar, [Formula: see text] and [Formula: see text] are two bounded invertible linear or conjugate linear operators.