Let be an infinite-dimensional separable Hilbert space. For a bounded linear operator on , recall that a closed subspace is a hyperinvariant subspace of if is invariant under any operator commuting with . We denote by Hlat the hyperinvariant subspace lattice of .
In the paper under review, the authors consider the class of cyclic quasianalytic contractions, and the subclass containing operators whose quasianalytic spectral sets are the unit circle. It is known by the work of the first author [J. Funct. Anal. 246, No. 2, 281–301 (2007; Zbl 1123.47008)] that every operator in has a rich invariant subspace lattice. The main result of the present paper asserts that for every operator , there exists an operator commuting with . It then follows that the identity Hlat Hlat holds. As a consequence, the Hyperinvariant Subspace Problem (HSP) in the class is equivalent to the HSP in the class .
The operator in the main theorem is given by , where is an appropriate -function on the unit disk. The existence of such an is proved by using tools from potential theory.