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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 $T \in {\cal L}(X)$.

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Linear Maps Preserving Matrices of Local Spectral Radius Zero at a Fixed Vector

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

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2 results

Linear Maps Preserving Matrices of Local Spectral Radius Zero at a Fixed Vector

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