  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.
