Maps preserving the local spectral radius zero of generalized product of operators

Abstract

ABSTRACTLet be the algebra of all bounded linear operators on a complex Banach space X. For an operator , let be the local spectral radius of T at any vector . For an integer , let be a finite sequence such that and at least one of the terms in appears exactly once. The generalized product of k operators is defined by and includes the usual product TS and the triple product TST. We show that a surjective map ϕ on satisfies for all and all if and only if there exists a map such that for all .