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Linear Surjective Maps Preserving at Least One Element from the Local Spectrum
Published online by Cambridge University Press: 23 January 2018
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 $T \in {\cal L}(X)$.
MSC classification
- 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