Lie product and local spectrum preservers

Abstract

Let X and Y be two infinite-dimensional complex Banach spaces, and fix two nonzero vectors x0∈X and y0∈Y. Let B(X) (resp. B(Y)) denote the algebra of all bounded linear operators on X (resp. on Y), and let Ex0(X) be the collection of all operators T∈B(X) for which x0 is an eigenvector and (T−r1X)2 is a nonzero scalar operator for some scalar r∈C. We show that a map φ from B(X) onto B(Y) satisfiesσφ(T)φ(S)−φ(S)φ(T)(y0)=σTS−ST(x0),(T,S∈B(X)), if and only if there are two functions η:B(X)→C and ξ:B(X)→{−1,1}, and a bijective bounded linear mapping A:X→Y such that Ax0=y0, the function ξ is constant on B(X)\\Ex0(X), andφ(T)=ξ(T)ATA−1+η(T)1Y for all T∈B(X).