On additive maps preserving certain semi-Fredholm subsets

Abstract

Let X and Y be two infinite dimensional real or complex Banach spaces, and let φ: ℒ(X) → ℒ(Y) be an additive surjective mapping that preserves semi-Fredholm operators in both directions. In the complex Hilbert space context, Mbekhta and Šemrl [M. Mbekhta and P. Šemrl, Linear maps preserving semi-Fredholm operators and generalized invertibility, Linear Multilinear Algebra 57 (2009), pp. 55–64] determined the structure of the induced map on the Calkin algebra. In this article, we show the following: given an integer n ≥ 1, if φ preserves in both directions ℳ n (X) (resp., 𝒬 n (X)), the set of semi-Fredholm operators on X of non-positive (resp., non-negative) index, having dimension of the kernel (resp., codimension of the range) less than n, then φ(T) = UTV for all T or φ(T) = UT*V for all T, where U and V are two bijective bounded linear, or conjugate linear, mappings between suitable spaces.