Maps preserving the local spectral subspace of skew-product of operators

Abstract

Let B(H) be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space H. For T ∈ B(H) and λ ∈ C, let HT ({λ}) denotes the local spectral subspace of T associated with {λ}. We prove that if ϕ : B(H) → B(H) be an additive map such that its range contains all operators of rank at most two and satisfiesfor all T, S ∈ B(H) and λ ∈ C, then there exist a unitary operator V in B(H) and a nonzero scalar µ such that ϕ(T ) = µT V * for all T ∈ B(H). We also show if ϕ1 and ϕ2 be additive maps from B(H) into B(H) such that their ranges contain all operators of rank at most two and satisfies) * ({λ}) = HT S * ({λ}) for all T, S ∈ B(H) and λ ∈ C. Then ϕ2(I) * is invertible, and ϕ1(T ) = T (ϕ2(I) * ) −1 and ϕ2(T ) = ϕ2(I) * T for all T ∈ B(H).