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Linear Surjective Maps Preserving at Least One Element from the Local Spectrum
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 61 / Issue 1 / February 2018
- Published online by Cambridge University Press:
- 23 January 2018, pp. 169-175
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- Article
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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 
$T \in {\cal L}(X)$.
