×

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

Citations:

Zbl 1123.47008
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andrievskii, V. V., Constructive function theory on sets of the complex plane through potential theory and geometric function theory, Surv. Approx. Theory, 2, 1-52 (2006) · Zbl 1113.30003
[2] Bercovici, H.; Foias, C.; Pearcy, C., On the hyperinvariant subspace problem, IV, Canad. J. Math., 60, 758-789 (2008) · Zbl 1153.47004
[3] Bercovici, H.; Kérchy, L., Spectral behaviour of \(C_{10}\)-contractions, (Operator Theory Live (2010), Theta: Theta Bucharest), 17-33 · Zbl 1222.47001
[4] Conway, J. B., A Course in Functional Analysis (1990), Springer-Verlag: Springer-Verlag New York · Zbl 0706.46003
[5] Foias, C.; Pearcy, C. M., (BCP)-operators and enrichment of invariant subspace lattices, J. Operator Theory, 9, 187-202 (1983) · Zbl 0533.47005
[6] Foias, C.; Pearcy, C. M.; Sz.-Nagy, B., Contractions with spectral radius one and invariant subspaces, Acta Sci. Math. (Szeged), 43, 273-280 (1981) · Zbl 0503.47004
[7] Garnett, J. B.; Marshall, D. E., Harmonic Measure, New Math. Monogr. (2005), Cambridge University Press: Cambridge University Press Cambridge, New York · Zbl 1077.31001
[8] Hoffman, K., Banach Spaces of Analytic Functions (1988), Dover Publications, Inc.: Dover Publications, Inc. New York · Zbl 0117.34001
[9] Kérchy, L., Isometric asymptotes of power bounded operators, Indiana Univ. Math. J., 38, 173-188 (1989) · Zbl 0693.47014
[10] Kérchy, L., On the hyperinvariant subspace problem for asymptotically nonvanishing contractions, Oper. Theory Adv. Appl., 127, 399-422 (2001) · Zbl 1008.47008
[11] Kérchy, L., Shift-type invariant subspaces of contractions, J. Funct. Anal., 246, 281-301 (2007) · Zbl 1123.47008
[12] Kérchy, L., Quasianalytic contractions and function algebras, Indiana Univ. Math. J., 60, 21-40 (2011) · Zbl 1275.47017
[13] Peherstorfer, F.; Steinbauer, R., Strong asymptotics of orthonormal polynomials with the aid of Greenʼs function, SIAM J. Math. Anal., 32, 385-402 (1999) · Zbl 0970.42017
[14] Pommerenke, Ch., Boundary Behaviour of Conformal Maps (1992), Springer-Verlag: Springer-Verlag New York · Zbl 0762.30001
[15] Ransford, T., Potential Theory in the Complex Plane (1995), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0828.31001
[16] Saff, E. B.; Totik, V., Logarithmic Potentials with External Fields, Grundlehren Math. Wiss., vol. 316 (1997), Springer-Verlag: Springer-Verlag Berlin, New York · Zbl 0881.31001
[17] Stahl, H.; Totik, V., General Orthogonal Polynomials, Encyclopedia Math. Appl., vol. 43 (1992), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0791.33009
[18] Sz.-Nagy, B.; Foias, C.; Bercovici, H.; Kérchy, L., Harmonic Analysis of Operators on Hilbert Space, Revised and Enlarged Edition, Universitext (2010), Springer-Verlag: Springer-Verlag New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.