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 $T \in B(H)$ and $λ\in \mathbb{C}$, let $H_{T}(\{λ\})$ denotes the local spectral subspace of $T$ associated with $\{λ\}$. We prove that if $φ:B(H)\rightarrow B(H)$ be an additive map such that its range contains all operators of rank at most two and satisfies $$H_{φ(T)φ(S)^{\ast}}(\{λ\})= H_{TS^{\ast}}(\{λ\})$$ for all $T, S \in B(H)$ and $λ\in \mathbb{C}$, then there exist a unitary operator $V$ in $B(H)$ and a nonzero scalar $μ$ such that $φ(T) = μTV^{\ast}$ for all $T \in B(H)$. We also show if $φ_{1}$ and $φ_{2}$ be additive maps from $B(H)$ into $B(H)$ such that their ranges contain all operators of rank at most two and satisfies $$H_{φ_{1}(T)φ_{2}(S)^{\ast}}(\{λ\})= H_{TS^{\ast}}(\{λ\})$$ for all $T, S \in B(H)$ and $λ\in \mathbb{C}$. Then $φ_{2}(I)^{\ast}$ is invertible, and $φ_{1}(T) = T(φ_{2}(I)^{\ast})^{-1}$ and $φ_{2}(T) =φ_{2}(I)^{\ast}T$ for all $T \in B(H)$.