ON 2-HYPONORMAL TOEPLITZ OPERATORS WITH FINITE RANK SELF-COMMUTATORS

Title & Authors
ON 2-HYPONORMAL TOEPLITZ OPERATORS WITH FINITE RANK SELF-COMMUTATORS
Kim, An-Hyun;

Abstract
Suppose $\small{T_{\varphi}}$ is a 2-hyponormal Toeplitz operator whose self-commutator has rank $\small{n{\geq}1}$. If $\small{H_{\bar{\varphi}}(ker[T^*_{\varphi},T_{\varphi}}$$\small{]}$$\small{)}$ contains a vector $\small{e_n}$ in a canonical orthonormal basis $\small{\{e_k\}_{k{\in}Z_+}}$ of $\small{H^2({\mathbb{T}})}$, then $\small{{\varphi}}$ should be an analytic function of the form ${\varphi} Keywords Toeplitz operators;finite rank self-commutators;subnormal;hyponormal;2-hyponormal; Language English Cited by References 1. M. B. Abrahamse, Subnormal Toeplitz operators and functions of bounded type, Duke Math. J. 43 (1976), no. 3, 597-604. 2. A. Aleman, Subnormal operators with compact selfcommutator, Manuscripta Math. 91 (1996), no. 3, 353-367. 3. I. Amemiya, T. Ito, and T. K. Wong, On quasinormal Toeplitz operators, Proc. Amer. Math. Soc. 50 (1975), 254-258. 4. J. Bram, Subnormal operators, Duke Math. J. 22 (1955), 75-94. 5. A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/1964), 89-102. 6. J. B. Conway, The Theory of Subnormal Operators, Math. Surveys and Monographs, 36, Amer. Math. Soc. Providence, 1991 7. C. Cowen, More subnormal Toeplitz operators, J. Reine Angew. Math. 367 (1986), 215-219. 8. C. Cowen, Hyponormal and subnormal Toeplitz operators, Surveys of Some Recent Results in Operator Theory, I (J. B. Conway and B. B. Morrel, eds.), Pitman Research Notes in Mathematics, pp. 155-167, Vol. 171, Longman, 1988. 9. C. Cowen, Hyponormality of Toeplitz operators, Proc. Amer. Math. Soc. 103 (1988), no. 3, 809-812. 10. C. Cowen and J. Long, Some subnormal Toeplitz operators, J. Reine Angew. Math. 351 (1984), 216-220. 11. R. E. Curto, S. H. Lee, and W. Y. Lee, Subnormality and 2-hyponormality for Toeplitz operators, Integral Equations Operator Theory 44 (2002), no. 2, 136-148. 12. R. E. Curto and W. Y. Lee, Subnormality and k-hyponormality of Toeplitz operators: A brief survey and open questions, Proceedings of Le Congres International des Mathema-tiques de Rabat, 73-81, (M. Mbekhta, ed.), The Theta Foundation, Bucharest, Romania, 2003. 13. P. Fan, Remarks on hyponormal trigonometric Toeplitz operators, Rocky Mountain J. Math. 13 (1983), no. 3, 489-493. 14. D. R. Farenick and W. Y. Lee, Hyponormality and spectra of Toeplitz operators, Trans. Amer. Math. Soc. 348 (1996), no. 10, 4153-4174. 15. C. Gu, A generalization of Cowen's characterization of hyponormal Toeplitz operators, J. Funct. Anal. 124 (1994), no. 1, 135-148. 16. T. Ito and T. K. Wong, Subnormality and quasinormality of Toeplitz operators, Proc. Amer. Math. Soc. 34 (1972), 157-164. 17. J. E. McCarthy and L. Yang, Subnormal operators and quadrature domains, Adv. Math. 127 (1997), no. 1, 52-72. 18. B. B. Morrel, A decomposition for some operators, Indiana Univ. Math. J. 23 (1973), 497-511. 19. T. Nakazi and K. Takahashi, Hyponormal Toeplitz operators and extremal problems of Hardy spaces, Trans. Amer. Math. Soc. 338 (1993), no. 2, 753-767. 20. D. Xia, Analytic theory of subnormal operators, Integral Equations Operator Theory 10 (1987), no. 6, 880-903. 21. D. Xia, On pure subnormal operators with finite rank self-commutators and related operator tuples, Integral Equations Operator Theory 24 (1996), no. 1, 106-125. 22. D. Yu, Hyponormal Toeplitz operators on$H^2(\mathbb{T})\$ with polynomial symbols, Nagoya Math. J. 144 (1996), 179-182.

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