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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.

47A15Invariant subspaces of linear operators
47A45Canonical models for contractions and nonselfadjoint operators
[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 · doi:emis:journals/SAT/papers/4/
[2]Bercovici, H.; Foias, C.; Pearcy, C.: On the hyperinvariant subspace problem, IV, Canad. J. Math. 60, 758-789 (2008) · Zbl 1153.47004 · doi:10.4153/CJM-2008-034-2
[3]Bercovici, H.; Kérchy, L.: Spectral behaviour of C10-contractions, , 17-33 (2010) · Zbl 1222.47001
[4]Conway, J. B.: A course in functional analysis, (1990)
[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)
[8]Hoffman, K.: Banach spaces of analytic functions, (1988) · Zbl 0734.46033
[9]Kérchy, L.: Isometric asymptotes of power bounded operators, Indiana univ. Math. J. 38, 173-188 (1989) · Zbl 0693.47014 · doi:10.1512/iumj.1989.38.38008
[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 · doi:10.1016/j.jfa.2007.01.011
[12]Kérchy, L.: Quasianalytic contractions and function algebras, Indiana univ. Math. J. 60, 21-40 (2011)
[13]Peherstorfer, F.; Steinbauer, R.: Strong asymptotics of orthonormal polynomials with the aid of Green’s function, SIAM J. Math. anal. 32, 385-402 (1999) · Zbl 0970.42017 · doi:10.1137/S0036141098343045
[14]Pommerenke, Ch.: Boundary behaviour of conformal maps, (1992) · Zbl 0762.30001
[15]Ransford, T.: Potential theory in the complex plane, (1995)
[16]Saff, E. B.; Totik, V.: Logarithmic potentials with external fields, Grundlehren math. Wiss. 316 (1997)
[17]Stahl, H.; Totik, V.: General orthogonal polynomials, Encyclopedia math. Appl. 43 (1992) · 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)