A Spectral Condition for Strong Compactness

Abstract

An algebra of bounded linear operators on a Banach space is said to be 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 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. A spectral condition is provided for an operator to be strongly compact and for its commutant to be a strongly compact algebra. This condition is applied to test strong compactness for some classes of operators, namely, bilateral weighted shifts, Cesaro operators, and composition operators. Some tools from function theory like the Mellin transform, the principle of analytic continuation, and the Weierstrass approximation theorem arise naturally in the pursuit of this program.