Journal of Operator Theory

Volume 72, Issue 2, Fall 2014  pp. 429-449.

Localizing algebras and invariant subspaces

Authors:  Miguel Lacruz (1) and Luis Rodriguez-Piazza (2)
Author institution: (1) Departamento de Analisis Matematico, Universidad de Sevilla, Apartado 1160, Sevilla 41080, Spain
(2) Departamento de Analisis Matematico, Universidad de Sevilla, Apartado 1160, Sevilla 41080, Spain

Summary:  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. Also, 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 $H^2({\mathbb D}).$ Finally, several conditions are provided for the algebra generated by a diagonal operator on a Hilbert space to be localizing.

DOI: http://dx.doi.org/10.7900/jot.2013may10.1995
Keywords:  localizing algebra, extended eigenvalue, invariant subspace