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Linear Surjective Maps Preserving at Least One Element from the Local Spectrum

Part of: Special classes of linear operators General theory of linear operators

Published online by Cambridge University Press:  23 January 2018

Constantin Costara
Constantin Costara*
Affiliation:
Faculty of Mathematics and Informatics, Ovidius University of Constanţa, Mamaia Boul. 124, 900527 Constanţa, Romania (cdcostara@univ-ovidius.ro)
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Abstract

Let X be a complex Banach space and denote by ${\cal L}(X)$ the Banach algebra of all bounded linear operators on X. We prove that if φ: ${\cal L}(X) \to {\cal L}(X)$ is a linear surjective map such that for each $T \in {\cal L}(X)$ and x ∈ X the local spectrum of φ(T) at x and the local spectrum of T at x are either both empty or have at least one common value, then φ(T) = T for all $T \in {\cal L}(X)$. If we suppose that φ always preserves the modulus of at least one element from the local spectrum, then there exists a unimodular complex constant c such that φ(T) = cT for all $T \in {\cal L}(X)$.

Keywords

linear preserverlocal spectrumlocal spectral radiusinner local spectral radius

MSC classification

Primary: 47B49: Transformers, preservers (operators on spaces of operators)
Secondary: 47A11: Local spectral properties
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 1 , February 2018 , pp. 169 - 175
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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