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Compression of quasianalytic spectral sets of cyclic contractions. (English) Zbl 1269.47009

Let $ℋ$ be an infinite-dimensional separable Hilbert space. For $T$ a bounded linear operator on $ℋ$, recall that a closed subspace $W\subset ℋ$ is a hyperinvariant subspace of $T$ if $W$ is invariant under any operator commuting with $T$. We denote by Hlat$\left(T\right)$ the hyperinvariant subspace lattice of $T$.

In the paper under review, the authors consider the class ${ℒ}_{0}\left(ℋ\right)$ of cyclic quasianalytic contractions, and the subclass ${ℒ}_{1}\left(ℋ\right)\subset {ℒ}_{0}\left(ℋ\right)$ 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 ${ℒ}_{1}\left(ℋ\right)$ has a rich invariant subspace lattice. The main result of the present paper asserts that for every operator $T\in {ℒ}_{0}\left(ℋ\right)$, there exists an operator ${T}_{1}\in {ℒ}_{1}\left(ℋ\right)$ commuting with $T$. It then follows that the identity Hlat$\left(T\right)=$ Hlat$\left({T}_{1}\right)$ holds. As a consequence, the Hyperinvariant Subspace Problem (HSP) in the class ${ℒ}_{0}\left(ℋ\right)$ is equivalent to the HSP in the class ${ℒ}_{1}\left(ℋ\right)$.

The operator ${T}_{1}$ in the main theorem is given by ${T}_{1}=f\left(T\right)$, where $f$ is an appropriate ${H}^{\infty }$-function on the unit disk. The existence of such an $f$ is proved by using tools from potential theory.

##### MSC:
 47A15 Invariant subspaces of linear operators 47A45 Canonical models for contractions and nonselfadjoint operators