# 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

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