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Proceedings of the American Mathematical Society

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Automatic continuity for linear surjective maps compressing the local spectrum at fixed vectors


Author: Constantin Costara
Journal: Proc. Amer. Math. Soc. 145 (2017), 2081-2087
MSC (2010): Primary 46H40; Secondary 47A11
DOI: https://doi.org/10.1090/proc/13364
Published electronically: October 26, 2016
MathSciNet review: 3611322
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Abstract: Let $X$ and $Y$ be complex Banach spaces and denote by $\mathcal {L}(X)$ and $\mathcal {L}(Y)$ the algebras of all bounded linear operators on $X$, respectively $Y$. Let also $x_0 \in X$ and $y_0 \in Y$ be nonzero vectors. We prove that if $\varphi : \mathcal {L}(X) \rightarrow \mathcal {L} (Y)$ is a linear surjective map such that for each $T \in \mathcal {L}(X)$ we have that the local spectrum of $\varphi (T)$ at $y_0$ is a subset of the local spectrum of $T$ at $x_0$, then $\varphi$ is automatically continuous. We also give a new proof for the automatic continuity of linear surjective maps decreasing the local spectral radius at some fixed nonzero vector. As a corollary, we obtain that the characterizations of J. Bračič and V. Müller for linear surjective maps on $\mathcal {L}(X)$ preserving the local spectrum/local spectral radius at some fixed vector can be obtained with no continuity assumptions on them.


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Additional Information

Constantin Costara
Affiliation: Faculty of Mathematics and Informatics, Ovidius University of Constanţa, Mamaia Boul. 124, 900527, Constanţa, Romania
MR Author ID: 676673
Email: cdcostara@univ-ovidius.ro

Received by editor(s): February 24, 2016
Received by editor(s) in revised form: June 28, 2016
Published electronically: October 26, 2016
Additional Notes: This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS - UEFISCDI, project number PN-II-RU-TE-2012-3-0042.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2016 American Mathematical Society