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

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 $\phi :B(H)\to B(H)$ be an additive map such that its range contains all operators of rank at most two and satisfies $$H_{\phi (T)\phi (S{)}^{\ast }}(\{\lambda \})=H_{T{S}^{\ast }}(\{\lambda \})$$ 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 $\phi (T)=\mu T{V}^{\ast }$ for all $T\in B(H)$. We also show if ${\phi }_1$ and ${\phi }_2$ be additive maps from $B(H)$ into $B(H)$ such that their ranges contain all operators of rank at most two and satisfies $$H_{{\phi }_1(T){\phi }_2(S{)}^{\ast }}(\{\lambda \})=H_{T{S}^{\ast }}(\{\lambda \})$$ for all $T,S\in B(H)$ and $\lambda \in \mathbb{C}$. Then ${\phi }_2(I{)}^{\ast }$ is invertible, and ${\phi }_1(T)=T({\phi }_2(I{)}^{\ast }{)}^{-1}$ and ${\phi }_2(T)={\phi }_2(I{)}^{\ast }T$ for all $T\in B(H)$.