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

TL;DR

This paper introduces a local spectral condition that guarantees strong compactness of bounded linear operators on Banach spaces and applies it to classify certain bilateral weighted shifts, extending previous results.

Contribution

It provides a new spectral criterion for strong compactness and applies it to characterize and construct examples of bilateral weighted shifts with specific properties.

Findings

A local spectral condition suffices for strong compactness.

Characterization of strongly compact bilateral weighted shifts.

Construction of invertible shifts with non-strongly compact inverses.

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.