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

   

 

A local spectral condition for strong compactness with some applications to bilateral weighted shifts


Authors: Miguel Lacruz and María del Pilar Romero de la Rosa
Journal: Proc. Amer. Math. Soc. 142 (2014), 243-249
MSC (2010): Primary 47B07; Secondary 47B37, 47L10
Published electronically: September 27, 2013
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Abstract: An algebra of bounded linear operators on a Banach space is said to be strongly compact provided that its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be strongly compact provided that the algebra with identity generated by the operator is strongly compact. Our interest in this notion stems from the work of Lomonosov on the existence of invariant subspaces. We consider a local spectral condition that is sufficient for a bounded linear operator on a Banach space to be strongly compact. This condition is then applied to describe a large class of strongly compact, injective bilateral weighted shifts on Hilbert spaces, extending earlier work of Fernández-Valles and the first author. Further applications are also derived; for instance, a strongly compact, invertible bilateral weighted shift is constructed in such a way that its inverse fails to be a strongly compact operator.


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Additional Information

Miguel Lacruz
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Avenida Reina Mercedes s/n, 41012 Sevilla, Spain
Email: lacruz@us.es

María del Pilar Romero de la Rosa
Affiliation: Departamento de Matemáticas, Universidad de Cádiz, Campus de Jerez, Avenida de la Universidad s/n, 11405 Jerez de la Frontera, Spain
Email: pilar.romero@uca.es

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11764-8
Received by editor(s): June 21, 2011
Received by editor(s) in revised form: March 5, 2012
Published electronically: September 27, 2013
Additional Notes: This research was partially supported by Ministerio de Ciencia e Innovación under Proyecto MTM2009-08934, and by Junta de Andalucía under Proyecto de Excelencia FQM 3737
Communicated by: Marius Junge
Article copyright: © Copyright 2013 American Mathematical Society