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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Strongly compact normal operators


Authors: Miguel Lacruz and Luis Rodríguez-Piazza
Journal: Proc. Amer. Math. Soc. 137 (2009), 2623-2630
MSC (2000): Primary 47B07, 47B15, 47L10
Published electronically: April 7, 2009
MathSciNet review: 2497474
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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.


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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
Email: piazza@us.es

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09927-4
PII: S 0002-9939(09)09927-4
Keywords: Algebras of operators on Hilbert spaces, strongly compact algebras, normal operators, Dirichlet algebras, strictly cyclic operators
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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.