Localizing algebras and diagonal operators

Abstract

The notion of a localizing algebra was introduced by Lomonosov, Radjavi, and Troitsky as a side condition to build invariant subspaces for operators on Banach spaces. The goal of this paper is to show that the unital algebra D generated by a single diagonal operator on a separable Hilbert space is localizing if and only if there is a non-zero compact operator in the weak closure of the unit ball of the algebra D:

Key words: Localizing algebra, diagonal operator, weak operator topology, bounded analytic function.