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

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

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

The operator T 1 in the main theorem is given by T 1 =f(T), where f is an appropriate H -function on the unit disk. The existence of such an f is proved by using tools from potential theory.

MSC:
47A15Invariant subspaces of linear operators
47A45Canonical models for contractions and nonselfadjoint operators
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