Linear Surjective Maps Preserving at Least One Element from the Local Spectrum
2018 ◽
Vol 61
(1)
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pp. 169-175
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AbstractLet 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 $T \in {\cal L}(X)$.
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2016 ◽
Vol 160
(3)
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pp. 413-421
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1969 ◽
Vol 21
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pp. 592-594
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2013 ◽
Vol 2013
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pp. 1-4
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1986 ◽
Vol 28
(1)
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pp. 69-72
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1978 ◽
Vol 30
(5)
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pp. 1045-1069
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1977 ◽
Vol 18
(2)
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pp. 197-198
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