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arXiv · 1308.4995

Localizing algebras and invariant subspaces

Miguel Lacruz·Luis Rodríguez-Piazza
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Abstract

It is shown that the algebra \(L^\infty(μ)\) of all bounded measurable functions with respect to a finite measure \(μ\) is localizing on the Hilbert space \(L^2(μ)\) if and only if the measure \(μ\) 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.

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Miguel Lacruz, Luis Rodríguez-Piazza. 2013-08-22. Localizing algebras and invariant subspaces. https://arxiv.org/abs/1308.4995

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