Nonlinear preservers of group invertible operators

Let B(X) be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space X. We prove that a bijective bicontinuous map Phi on B(X) preserves the difference of group invertible operators in both directions if and only if F is either of the form Phi(T) = alpha ATA(-1) + Phi(0) or of the form Phi(T) = alpha BT*B-1 + Phi(0), where a is a nonzero scalar, A : X -> X and B : X* -> X are two bounded invertible linear or conjugate linear operators.