This is an announcement for the paper "A local spectral condition for strong compactness with some applications to bilateral weighted shifts" by Miguel Lacruz and Maria del Pilar Romero de la Rosa.
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.
Archive classification: math.FA math.OA
Mathematics Subject Classification: 47B07
Remarks: 7 pages, to appear in Proc. Amer. Math. Soc
Submitted from: lacruz@us.es
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1208.3245
or