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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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Strongly compact normal operators
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by Miguel Lacruz and Luis Rodríguez-Piazza PDF
Proc. Amer. Math. Soc. 137 (2009), 2623-2630 Request permission

Abstract:

An algebra of bounded linear operators on a Hilbert space is said to be strongly compact if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Hilbert space is said to be strongly compact if the unital algebra generated by the operator is strongly compact. We show that the position operator on the space of square integrable functions with respect to a finite measure of compact support is strongly compact if and only if the restriction of the measure to the boundary of the polynomially convex hull of its support is purely atomic. This result is applied to construct a strongly compact operator that generates a weakly closed unital algebra that fails to be strongly compact. Also, we construct an operator such that the weakly closed unital algebra generated by the operator is strongly compact but the bicommutant of the operator fails to be a strongly compact algebra. Finally, we prove that a strongly compact operator cannot be strictly cyclic.
References
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Additional Information
  • Miguel Lacruz
  • Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Apartado de Correos 1160, 41080 Sevilla, Spain
  • Email: lacruz@us.es
  • Luis Rodríguez-Piazza
  • Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Apartado de Correos 1160, 41080 Sevilla, Spain
  • MR Author ID: 245308
  • Email: piazza@us.es
  • Received by editor(s): July 4, 2008
  • Published electronically: April 7, 2009
  • Additional Notes: The first author’s research was partially supported by Junta de Andalucía under Grant FQM-260.
    The second author’s research was partially supported by Junta de Andalucía under Grant FQM-627.
  • Communicated by: Marius Junge
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 2623-2630
  • MSC (2000): Primary 47B07, 47B15, 47L10
  • DOI: https://doi.org/10.1090/S0002-9939-09-09927-4
  • MathSciNet review: 2497474