zbMATH — the first resource for mathematics

Subnormal and quasinormal Toeplitz operators with matrix-valued rational symbols. (English) Zbl 1318.47031
In [Pac. J. Math. 223, No. 1, 95–111 (2006; Zbl 1125.47019)], C.-X. Gu et al. extended Cowen’s theorem to block Toeplitz operators: For each \(\Phi \in L^\infty_{M_n}\), let \(\mathcal{E}(\Phi):= \{ K \in H^\infty_{M_n} : \|K\|_\infty \leq 1 ~\text{and}~ \Phi- K\Phi^* \in H^\infty_{M_n} \}\). Then \(T_\Phi\) is hyponormal if and only if \(\Phi\) is normal and \(\mathcal{E}(\Phi)\) is nonempty. First, the authors establish M. B. Abrahamse’s theorem [Duke Math. J. 43, 597–604 (1976; Zbl 0332.47017)] for matrix-valued rational symbols. Let \(\Phi \in L^\infty_{M_n}\) be a matrix-valued rational function having a “matrix pole”, i.e., there exists \(\alpha \in \mathbb{D}\) for which \(\ker H_\Phi \subseteq (z- \alpha)H^2_{\mathbb{C}^n}\), where \(H_\Phi\) denotes the Hankel operator with symbol \(\Phi\). Then the authors prove that, if (i) \(T_\Phi\) is hyponormal and (ii) \(\ker[T_\Phi^*, T_\Phi]\) is invariant for \(T_\Phi\), then \(T_\Phi\) is normal. Hence, in particular, if \(T_\Phi\) is subnormal, then \(T_\Phi\) is normal. Next, the authors establish Amemiya-Ito-Wong’s theorem [I. Amemiya et al., Proc. Am. Math. Soc. 50, 254–258 (1975; Zbl 0339.47019)] for matrix-valued rational symbols. They prove that every pure quasinormal Toeplitz operator with a matrix-valued rational symbol is unitarily equivalent to an analytic Toeplitz operator.

47B20 Subnormal operators, hyponormal operators, etc.
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
15A83 Matrix completion problems
30H10 Hardy spaces
47A20 Dilations, extensions, compressions of linear operators
Full Text: DOI
[1] Abdollahi, A., Self-commutators of automatic composition operators on the Dirichlet space, Proc. Amer. Math. Soc., 136, 9, 3185-3193, (2008) · Zbl 1151.47032
[2] Abrahamse, M. B., Subnormal Toeplitz operators and functions of bounded type, Duke Math. J., 43, 597-604, (1976) · Zbl 0332.47017
[3] Amemiya, I.; Ito, T.; Wong, T. K., On quasinormal Toeplitz operators, Proc. Amer. Math. Soc., 50, 254-258, (1975) · Zbl 0339.47019
[4] Böttcher, A.; Silbermann, B., Analysis of Toeplitz operators, (2006), Springer Berlin-Heidelberg
[5] Bram, J., Subnormal operators, Duke Math. J., 22, 75-94, (1955) · Zbl 0064.11603
[6] Conway, J. B., The theory of subnormal operators, Math. Surveys Monogr., vol. 36, (1991), Amer. Math. Soc. Providence, RI · Zbl 0743.47012
[7] Cowen, C., More subnormal Toeplitz operators, J. Reine Angew. Math., 367, 215-219, (1986) · Zbl 0577.47025
[8] Cowen, C., Hyponormality of Toeplitz operators, Proc. Amer. Math. Soc., 103, 809-812, (1988) · Zbl 0668.47021
[9] Cowen, C.; Long, J., Some subnormal Toeplitz operators, J. Reine Angew. Math., 351, 216-220, (1984) · Zbl 0532.47019
[10] Curto, R. E.; Hwang, I. S.; Lee, W. Y., Hyponormality and subnormality of block Toeplitz operators, Adv. Math., 230, 2094-2151, (2012) · Zbl 1248.47029
[11] Curto, R. E.; Hwang, I. S.; Lee, W. Y., Which subnormal Toeplitz operators are either normal or analytic?, J. Funct. Anal., 263, 8, 2333-2354, (2012) · Zbl 1263.47033
[12] Curto, R. E.; Lee, W. Y., Joint hyponormality of Toeplitz pairs, Mem. Amer. Math. Soc., vol. 712, (2001), Amer. Math. Soc. Providence · Zbl 0982.47022
[13] Curto, R. E.; Lee, W. Y., Towards a model theory for 2-hyponormal operators, Integral Equations Operator Theory, 44, 290-315, (2002) · Zbl 1052.47016
[14] Curto, R. E.; Lee, W. Y., Subnormality and k-hyponormality of Toeplitz operators: A brief survey and open questions, (Operator Theory and Banach Algebras, Rabat, 1999, (2003), Theta Bucharest), 73-81 · Zbl 1108.47300
[15] Douglas, R. G., Banach algebra techniques in operator theory, (1972), Academic Press New York · Zbl 0247.47001
[16] Douglas, R. G., Banach algebra techniques in the theory of Toeplitz operators, CBMS Reg. Conf. Ser. Math., vol. 15, (1973), Amer. Math. Soc. Providence · Zbl 0252.47025
[17] Foiaş, C.; Frazho, A., The commutant lifting approach to interpolation problems, Oper. Theory Adv. Appl., vol. 44, (1993), Birkhäuser Boston
[18] Gu, C.; Hendricks, J.; Rutherford, D., Hyponormality of block Toeplitz operators, Pacific J. Math., 223, 95-111, (2006) · Zbl 1125.47019
[19] Halmos, P. R., Ten problems in Hilbert space, Bull. Amer. Math. Soc., 76, 887-933, (1970) · Zbl 0204.15001
[20] Halmos, P. R., Ten years in Hilbert space, Integral Equations Operator Theory, 2, 529-564, (1979) · Zbl 0429.47001
[21] Halmos, P. R., A Hilbert space problem book, (1982), Springer New York · Zbl 0144.38704
[22] Ito, T.; Wong, T. K., Subnormality and quasinormality of Toeplitz operators, Proc. Amer. Math. Soc., 34, 157-164, (1972) · Zbl 0239.47027
[23] MacCluer, B. D.; Pons, M. A., Automatic composition operators on Hardy and Bergman spaces in the ball, Houston J. Math., 32, 4, 1121-1132, (2006) · Zbl 1113.47016
[24] Morrel, B. B., A decomposition for some operators, Indiana Univ. Math. J., 23, 497-511, (1973) · Zbl 0252.47015
[25] Nakazi, T.; Takahashi, K., Hyponormal Toeplitz operators and extremal problems of Hardy spaces, Trans. Amer. Math. Soc., 338, 753-769, (1993) · Zbl 0798.47018
[26] Nikolskii, N. K., Treatise on the shift operator, (1986), Springer New York
[27] Peller, V. V., Hankel operators and their applications, (2003), Springer New York · Zbl 1030.47002
[28] Potapov, V. P., On the multiplicative structure of J-nonexpansive matrix functions, Tr. Mosk. Mat. Obs., Amer. Math. Soc. Transl. Ser. 2, 15, 131-243, (1966), (in Russian); English transl. in: · Zbl 0090.05403
[29] Yakubovich, D. V., Real separated algebraic curves, quadrature domains, Ahlfors type functions and operator theory, J. Funct. Anal., 236, 25-58, (2006) · Zbl 1131.47062
[30] Zorboska, N., Closed range essentially normal composition operators, Acta Sci. Math. (Szeged), 65, 287-292, (1999) · Zbl 0938.47022
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.