A local spectral condition for strong compactness with some applications to bilateral weighted shifts
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- by Miguel Lacruz and María del Pilar Romero de la Rosa PDF
- Proc. Amer. Math. Soc. 142 (2014), 243-249 Request permission
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.References
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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
- 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
- © Copyright 2013 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 243-249
- MSC (2010): Primary 47B07; Secondary 47B37, 47L10
- DOI: https://doi.org/10.1090/S0002-9939-2013-11764-8
- MathSciNet review: 3119199