Essentially Commuting Dual Truncated Toeplitz Operators

Abstract

In this paper, the authors completely characterize when two dual truncated Toeplitz operators are essentially commuting and when the semicommutator of two dual truncated Toeplitz operators is compact. Their main idea is to study dual truncated Toeplitz operators via Hankel operators, Toeplitz operators and function algebras.

Keywords

Hardy space / Dual truncated Toeplitz operator / Essentially commuting

References

[1] Axler S, Chang S-Y A, Sarason D. Products of Toeplitz operators. Integral Equations Operator Theory, 1978, 1(3): 285-309

[2] Axler S, Čučkvoić Ž. Commuting Toeplitz operators with harmonic symbols. Integral Equations Operator Theory, 1991, 14(1): 1-12

[3] Barría J, Halmos P R. Asymptotic Toeplitz operators. Trans. Amer. Math. Soc., 1982, 273(2): 621-630

[4] Brown A, Halmos P R. Algebraic properties of Toeplitz operators. J. Reine. Angew. Math., 1964, 213(1): 89-102

[5] Câmara C, Kliś-Garlicka K, Łanucha B, Ptak M. Invertibility, Fredholmness and kernels of dual truncated Toeplitz operators. Banach J. Math. Anal., 2020, 14(4): 1558-1580

[6] Ding X, Sang Y. Dual truncated Toeplitz operators. J. Math. Anal. Appl., 2018, 461(1): 929-946

[7] Ding X, Sun S, Zheng D. Commuting Toeplitz operators on the bidisk. J. Funct. Anal., 2012, 263(11): 3333-3357

[8] Douglas R. Banach Algebra Techniques in Operator Theory, 1998 2 New York: Springer-Verlag

[9] Duren P L. Theory of H ^p Spaces, 2000, New York: Academic Press

[10] Garcia S R, Mashreghi J, Ross W T. Introduction to Model Spaces and Their Operators, 2016, Cambridge: Cambridge University Press

[11] Garnett J B. Bounded Analytic Functions, 1981, New York: Academic Press

[12] Gorkin P, Zheng D. Essentially commuting Toeplitz operators. Pacific J. Math., 1999, 190(1): 87-109

[13] Gu C, Zheng D. The semicommutator of Toeplitz operators on the bidisc. J. Operator Theory, 1997, 38(1): 173-193

[14] Guo K, Zheng D. Essentially commuting Hankel and Toeplitz operators. J. Funct. Anal., 2003, 201(1): 121-147

[15] Guo K, Zheng D. The distribution function inequality for a finite sum of finite products of Toeplitz operators. J. Funct. Anal., 2005, 218(1): 1-53

[16] Guo K, Sun S, Zheng D. Finite rank commutators and semicommutators of Toeplitz operators with harmonic symbols. Illinois J. Math., 2007, 51(2): 583-596

[17] Hoffman K. Banach Spaces of Analytic Functions, 1962, Englewood Cliffs, NJ: Prentice-Hall, Inc.

[18] Sang Y, Qin Y, Ding X. Dual truncated Toeplitz C*-algebras. Banach J. Math. Anal., 2019, 13(2): 275-292

[19] Sang Y, Qin Y, Ding X. A theorem of Brown-Halmos type for dual truncated Toeplitz operators. Ann. Funct. Anal., 2020, 11(1): 271-284

[20] Stroethoff K. Essentially commuting Toeplitz operators with harmonic symbols. Canad. J. Math., 1993, 45(5): 1080-1093

[21] Stroethoff K, Zheng D. Products of Hankel and Toeplitz operators on the Bergman space. J. Funct. Anal., 1999, 169(1): 289-313

[22] Stroethoff K, Zheng D. Algebraic and spectral properties of dual Toeplitz operators. Trans. Amer. Math. Soc., 2002, 354(6): 2495-2520

[23] Volberg A. Two remarks concerning the theorem of S. Axler., S.-Y. A. Chang and Sarason, D.. J. Operator Theory, 1982, 7(2): 209-218

[24] Zheng D. The distribution function inequality and products of Toeplitz operators and Hankel operators. J. Funct. Anal., 1996, 138(2): 447-501

[25] Zheng D. Toeplitz operators and Hankel operators on the Hardy space of the unit sphere. J. Funct. Anal., 1997, 149(1): 1-24

[26] Zhu K. Operator Theory in Function Spaces, 2007 2 Providence, RI: American Mathematical Society