Operator products and algebraic spectral subspace preservers

Abstract

Let Ҳ be an infinite-dimensional complex Banach space and ₿(Ҳ) be the algebra of all bounded linear operators on Ҳ. For T ∈ ₿(Ҳ), and a fixed nonzero complex scalar λ0, we denote by E_T ({λ0}), the algebraic spectral subspace of T associated with {λ0}. In this paper, we characterize maps ϕ on ₿(Ҳ) for which whose ranges contain all operators of rank at most two (resp. at most four), and that satisfy E_TS({λ0}) = E_ϕ(T)ϕ(S)({λ0}) (resp. E_TST ({λ0}) = E_{ϕ(T)ϕ(S)ϕ(T)}({λ0})), for all T, S ∈ ₿(Ҳ).