Localizing algebras and invariant subspaces

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.