On contractions with compact defects

M. F. Gamal'

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 37, Tome 366 (2009), pp. 13-41

Résumé

In [8], the following question was posed: suppose that $T$ is a contraction of class $C_{10}$ such that $I-T^\ast T$ is compact and the spectrum of $T$ is the unit disk. Can the isometric asymptote of $T$ be a reductive unitary operator? In this paper, we give a positive answer to this question. We construct two kinds of examples. One of them is the operators of multiplication by the independent variable in the closure of analytic polynomials in $L^2(\nu)$, where $\nu$ is an appropiate positive finite Borel measure on the closed unit disk. The second kind of examples is based on Theorem 6.2 in [5]. We obtain a contraction $T$ satisfying all required conditions and such that $I-T^\ast T$ belongs to Schatten–von Neumann classes $\mathfrak S_p$ for all $p>1$. Also we give an example of a contraction $T$ such that $I-T^\ast T$ belongs to $\mathfrak S_p$ for all $p>1$, $T$ is quasisimilar to a unitary operator and has “more” invariant subspaces than this unitary operator. Also, following [2], we show that if a subset of the unit circle is the spectrum of a contraction quasisimilar to an absolutely continuous unitary operator, then this contraction $T$ can be chosen such that $I-T^\ast T$ is compact. Bibl. – 29 titles.

Bibliographie

[1] B. Beauzamy, Introduction to operator theory and invariant subspaces, North Holland, Amsterdam, 1988 | MR | Zbl

[2] H. Bercovici, L. Kérchy, “On the spectra of $C_{11}$-contractions”, Proc. Amer. Math. Soc., 95 (1985), 412–418 | DOI | MR | Zbl

[3] L. de Branges, “Some Hilbert spaces of analitic functions. II”, J. Math. Anal. Appl., 11 (1965), 44–72 | DOI | MR | Zbl

[4] B. Chevreau, “Sur les contractions à calcul fonctionnel isométrique. II”, J. Operator Theory, 20 (1988), 269–293 | MR | Zbl

[5] B. Chevreau, G. Exner, C. Pearcy, “On the structure of contraction operators. III”, Michigan Math. J., 36 (1989), 29–62 | DOI | MR | Zbl

[6] I. Colojoara, C. Foiaş, Theory of generalized spectral operators, Gordon and Breach, Science Publishers, 1968 | MR | Zbl

[7] J. B. Conway, The theory of subnormal operators, Math. Surveys and Monographs, 36, Amer. Math. Soc., 1991 | DOI | MR | Zbl

[8] B. P. Duggal, S. V. Djordjević, C. S. Kubrusly, “Hereditarily normaloid contractions”, Acta Sci. Math. (Szeged), 71 (2005), 337–352 | MR | Zbl

[9] J. Esterle, “Singular inner functions and biinvariant subspaces for dissymetric weighted shifts”, J. Funct. Anal., 144 (1997), 64–104 | DOI | MR | Zbl

[10] M. F. Gamal, “$C_0$-szhatiya: zhordanova model i reshetki invariantnykh podprostranstv”, Algebra i analiz, 15:5 (2003), 198–227 | MR | Zbl

[11] M. F. Gamal', “On contractions that are quasiaffine transforms of unilateral shifts”, Acta Sci. Math. (Szeged), 74 (2008), 755–765 | MR

[12] Dzh. Garnett, Ogranichennye analiticheskie funktsii, Mir, M., 1984 | MR

[13] D. A. Herrero, “Indecomposable compact perturbations of the bilateral shift”, Proc. Amer. Math. Soc., 62 (1977), 254–258 | DOI | MR | Zbl

[14] D. A. Herrero, “On multicyclic operators”, Integral Equations and Operator Theory, 1 (1978), 57–102 | DOI | MR | Zbl

[15] L. Kérchy, “A description of invariant subspaces of $C_{11}$-contractions”, J. Operator Theory, 15 (1986), 327–344 | MR | Zbl

[16] L. Kérchy, “Contraction being weakly similar to unitaries”, Oper. Theory Adv. Appl., 17 (1986), 187–200 | MR | Zbl

[17] L. Kérchy, “On the spectra of contractions belonging to special classes”, J. Funct. Anal., 67 (1986), 153–166 | DOI | MR | Zbl

[18] L. Kérchy, “Isometric asymptotes of power bounded operators”, Indiana Univ. Math. J., 38 (1989), 173–188 | DOI | MR | Zbl

[19] L. Kérchy, “On the hyperinvariant subspace problem for asymptotically nonvanishing contractions”, Oper. Theory Adv. Appl., 127 (2001), 399–422 | MR | Zbl

[20] L. Kérchy, “Shift-type invariant subspaces of contractions”, J. Funct. Anal., 246 (2007), 281–301 | DOI | MR | Zbl

[21] S. V. Khruschev, “Problema odnovremennoi approksimatsii i stiranie osobennostei integralov tipa Koshi”, Trudy MIAN, 130, 1978, 124–195

[22] T. L. Kriete, D. Trutt, “The Cesàro operator in $l^2$ is subnormal”, Amer. J. Math., 93 (1971), 215–225 | DOI | MR | Zbl

[23] T. L. Kriete, D. Trutt, “On the Cesàro operator”, Indiana Univ. Math. J., 24 (1974), 197–214 | DOI | MR | Zbl

[24] M. Radjabalipour, H. Radjavi, “On invariant subspaces of compact perturbations of operators”, Rev. Roumaine Math. Pures Appl., 21 (1976), 1247–1260 | MR | Zbl

[25] L. A. Sakhnovich, “Operatory, podobnye unitarnym, s absolyutno nepreryvnym spektrom”, Funkts. analiz i ego prilozheniya, 2:1 (1968), 51–63 | MR | Zbl

[26] B. Sekefalvi-Nad, Ch. Foyash, Garmonicheskii analiz operatorov v gilbertovom prostranstve, Mir, M., 1970 | MR

[27] K. Takahashi, “$C_{1\cdot}$-contractions with Hilbert–Schmidt defect operators”, J. Operator Theory, 12 (1984), 331–347 | MR | Zbl

[28] M. Uchiyama, “Contractions and unilateral shifts”, Acta Sci. Math. (Szeged), 46 (1983), 345–356 | MR | Zbl

[29] M. Uchiyama, “Contractions with $(\sigma,\mathrm c)$ defect operators”, J. Operator Theory, 12 (1984), 221–233 | MR | Zbl