Miguel Lacruz

Strongly compact normal operators

An algebra of bounded linear operators on a Hilbert 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 Hilbert space is said to be strongly compact if the unital algebra generated by the operator is strongly compact. We show that the position operator on the space of square integrable functions with respect to a finite measure of compact support is strongly compact if and only if the restriction of the measure to the boundary of the polynomially convex hull of its support is purely atomic. This result is applied to construct a strongly compact operator that generates a weakly closed unital algebra that fails to be strongly compact. Also, we construct an operator such that the weakly closed unital algebra generated by the operator is strongly compact but the bicommutant of the operator fails to be a strongly compact algebra. Finally, we prove that a strongly compact operator cannot be strictly cyclic.

Essential Norm of Composition Operators on Banach Spaces of Hölder Functions

Let be a pointed compact metric space, let , and let be a base point preserving Lipschitz map. We prove that the essential norm of the composition operator induced by the symbol on the spaces and is given by the formula whenever the dual space has the approximation property. This happens in particular when is an infinite compact subset of a finite-dimensional normed linear space.

A note on transitive localizing algebras

A simple proof is provided for a theorem of Troitsky that every nonzero quasinilpotent operator on a Banach space whose commutant is a localizing algebra has a nontrivial hyperinvariant subspace.

Applications of fixed point theorems in the theory of invariant subspaces

We survey several applications of fixed point theorems in the theory of invariant subspaces. The general idea is that a fixed point theorem applied to a suitable map yields the existence of invariant subspaces for an operator on a Banach space.MSC:47A15, 47H10.