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Zbl pre06104942
Kérchy, László; Totik, Vilmos
Compression of quasianalytic spectral sets of cyclic contractions. (English)
[J] J. Funct. Anal. 263, No. 9, 2754-2769 (2012). ISSN 0022-1236

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 $T$. \par 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 current 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 $\mathcal{L}_1(\mathcal{H})$. \par 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.
[Trieu Le (Toledo)]

MSC 2000:: *47A15 Invariant subspaces of linear operators
47A45 Canonical models for contractions and nonselfadjoint operators

Keywords: quasianalytic spectral set; hyperinvariant subspace; equilibrium measure; absolute continuity

Citations: Zbl 1123.47008

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