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Compression of quasianalytic spectral sets of cyclic contractions. (English) Zbl 1269.47009
Let $$\mathcal{H}$$ be an infinite-dimensional separable Hilbert space. For $$T$$ a bounded linear operator on $$\mathcal{H}$$, recall that a closed subspace $$W\subset\mathcal{H}$$ 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 $$\mathcal{L}_0(\mathcal{H})$$ of cyclic quasianalytic contractions, and the subclass $$\mathcal{L}_{1}(\mathcal{H})\subset\mathcal{L}_0(\mathcal{H})$$ 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 $$\mathcal{L}_{1}(\mathcal{H})$$ has a rich invariant subspace lattice. The main result of the present paper asserts that for every operator $$T\in\mathcal{L}_0(\mathcal{H})$$, there exists an operator $$T_1\in\mathcal{L}_1(\mathcal{H})$$ 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 $$\mathcal{L}_0(\mathcal{H})$$ is equivalent to the HSP in the class $$\mathcal{L}_1(\mathcal{H})$$.
The operator $$T_1$$ in the main theorem is given by $$T_1=f(T)$$, 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.
Reviewer: Trieu Le (Toledo)

##### MSC:
 47A15 Invariant subspaces of linear operators 47A45 Canonical models for contractions and nonselfadjoint linear operators
Zbl 1123.47008
Full Text:
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