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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Automatic continuity for linear surjective maps compressing the local spectrum at fixed vectors
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by Constantin Costara PDF
Proc. Amer. Math. Soc. 145 (2017), 2081-2087 Request permission

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
  • © Copyright 2016 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 2081-2087
  • MSC (2010): Primary 46H40; Secondary 47A11
  • DOI: https://doi.org/10.1090/proc/13364
  • MathSciNet review: 3611322