Localizing algebras and invariant subspaces

TL;DR

The paper characterizes when certain algebras of functions are localizing on Hilbert spaces, explores conditions for diagonal operator algebras to be localizing, and extends results on hyperinvariant subspaces for operators with localizing extended eigenoperators.

Contribution

It provides a complete characterization of when the algebra of bounded measurable functions is localizing, analyzes the non-localizing nature of bounded analytic multipliers, and extends existing theorems on hyperinvariant subspaces.

Findings

L^() is localizing iff  has an atom.

H^() is not localizing on Bergman and Hardy spaces.

Conditions for diagonal operator algebras to be localizing.

Abstract

It is shown that the algebra \(L^\infty(\mu)\) of all bounded measurable functions with respect to a finite measure \(\mu\) is localizing on the Hilbert space \(L^2(\mu)\) if and only if the measure \(\mu\) has an atom. Next, it is shown that the algebra \(H^\infty({\mathbb D})\) of all bounded analytic multipliers on the unit disc fails to be localizing, both on the Bergman space \(A^2({\mathbb D})\) and on the Hardy space \(H^2({\mathbb D}).\) Then, several conditions are provided for the algebra generated by a diagonal operator on a Hilbert space to be localizing. Finally, a theorem is provided about the existence of hyperinvariant subspaces for operators with a localizing subspace of extended eigenoperators. This theorem extends and unifies some previously known results of Scott Brown and Kim, Moore and Pearcy, and Lomonosov, Radjavi and Troitsky.