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