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arXiv · 1208.3245

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

Miguel Lacruz·Maria del Pilar Romero de la Rosa
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Abstract

An algebra of bounded linear operators on a Banach space is said to be {\em strongly compact} if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be {\em strongly compact} if 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 provide 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 Fern\'andez-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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Miguel Lacruz, Maria del Pilar Romero de la Rosa. 2012-08-15. A local spectral condition for strong compactness with some applications to bilateral weighted shifts. https://arxiv.org/abs/1208.3245

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