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

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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)… Click to show full abstract

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: spectrum; element local; one element; least one; local spectrum; linear surjective

Journal Title: Proceedings of the Edinburgh Mathematical Society
Year Published: 2018

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