Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46H40, 47A11

Retrieve articles in all journals with MSC (2010): 46H40, 47A11


Additional Information

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

DOI: https://doi.org/10.1090/proc/13364
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

American Mathematical Society