On Additive Maps Preserving the Local Spectral Subspace of‎ ‎triple product of operator‎

Articles in Press

Document Type : Original Article

Authors
Elahe M‎. ‎Hossienpour 1
Rohollah Parvinianzadeh 2
Meysam Asadipour 1
1 Department of Mathematics‎, ‎University of Yasouj‎, ‎Yasouj‎, ‎Iran
2 Department of Mathematics Assistant Professor of Mathematics Yasouj University Yasouj, Iran
‎10.22034/maco.6.1.7
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 $\varphi_{1}$ and $\varphi_{2}$ be additive maps from $B(H)$ into $B(H)$ such that their ranges contain all operators of rank at most‎ ‎four and satisfies‎
‎$$H_{\varphi_{1}(T)\varphi_{2}(S)^{\ast}\varphi_{1}(T)}(\{\lambda\})= H_{TS^{\ast}T}(\{\lambda\})$$‎
‎for all $T‎, ‎S \in B(H)$ and $\lambda \in \mathbb{C}$‎. ‎Then there exist two invertible operators $P:H \longrightarrow H$ and $Q:H \longrightarrow H$ such that $\varphi_{1}(T) =PTP$ and $\varphi_{2}(T) =QTQ$ for all $T \in B(H)$‎.
Keywords
Local spectrum
Local spectral subspace
Nonlinear preservers
Rank-one operators
Subjects
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Mathematical Analysis and Convex Optimization

Articles in Press, Accepted Manuscript
Available Online from 12 September 2026