Some results on local spectral subspace preserving maps

Arvinianzadeh, Rohollah; Mohammadhossienpour, ٍElahe

doi:10.22055/jamm.2026.51710.2466

Some results on local spectral subspace preserving maps

Document Type : Original Paper

Authors

Rohollah arvinianzadeh ¹

ٍElahe Mohammadhossienpour ²

¹ Department of Mathematics, Faculty of Basic Science,, Yasuj University, Yasuj,, Iran

² Department of Mathematics‎, ‎Faculty of Basic Science‎, ‎Yasuj University‎, ‎Yasuj‎, ‎Iran‎

10.22055/jamm.2026.51710.2466

Abstract

‎Let‎ $B(X)$‎ ‎be the algebra of all bounded linear operators on infinite-dimensional complex Banachspace $X$.For‎

‎$T \in B(X)$‎ ‎and‎ ‎$\lambda \in \mathbb{C}$‎, ‎let‎ ‎$X_T(\{\lambda\})$‎ ‎denotes the local spectral subspace of $T$ associated with‎ ‎$\{\lambda\}$‎. ‎We show that if two additive maps‎ ‎$\varphi_1$‎ ‎and‎ ‎$\varphi_2$‎ ‎from‎ ‎$B(X)$‎ ‎onto‎ ‎$B(X)$‎ ‎satisfy‎ X_{\varphi_1(T)\varphi_2(S)}(\{\lambda\}) = X_{T S}(\{\lambda\})‎ ‎$

for all‎ ‎$T,S \in B(X)$‎ ‎and‎ ‎$\lambda \in \mathbb{C}$‎, ‎then there exists a bijective linear map‎ ‎$P‎ : ‎X \to X$‎

‎such that‎ ‎$\varphi_1(T) = T P^{-1}$‎ ‎and‎ ‎$\varphi_2(T) = P T$‎ ‎for all‎ ‎$T \in B(X)$‎.

‎Also‎, ‎we obtain some interesting results in this direction‎.

Keywords

‎‎Local spectral subspace‎

Local spectrum‎‎‎

‎Nonlinear preservers‎

‎Finite‎ rank ‎operator

‎ ‎Rank-one ‎operators‎

Subjects