  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.
