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Essentially commuting Hankel and Toeplitz operators. (English) Zbl 1036.47016
Let \(T_f\) and \(H_g\) denote Toeplitz and Hankel operators in the Hardy space \(H^2\) which is a subspace of \(L^2 (\partial D, d\sigma),\) where \(d\sigma\) is the normalized Lebesgue measure on the unit circle \(\partial D\). The authors study the question when the commutator \([H_g, T_f]=H_gT_f-T_fH_g\) of Hankel and Toeplitz operators is compact. They find some criteria for \([H_g, T_f]\) to be compact.

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
Full Text: DOI
[1] Axler, S.; Chang, S.-Y.A.; Sarason, D., Product of Toeplitz operators, Integral equations operator theory, 1, 285-309, (1978) · Zbl 0396.47017
[2] Barrı́a, J., On Hankel operators not in the Toeplitz algebra, Proc. amer. math. soc., 124, 1507-1511, (1996) · Zbl 0845.47017
[3] Barrı́a, J.; Halmos, P., Asymptotic Toeplitz operators, Trans. amer. math. soc., 273, 621-630, (1982) · Zbl 0522.47020
[4] Böttcher, A.; Silbermann, B., Analysis of Toeplitz operators, (1990), Springer Berlin · Zbl 0721.47028
[5] Carleson, L., Interpolations by bounded analytic functions and the corona problem, Ann. math., 76, 547-559, (1962) · Zbl 0112.29702
[6] Chang, S.-Y.A., A characterization of Douglas subalgebras, Acta math., 137, 81-89, (1976) · Zbl 0332.46035
[7] Douglas, R.G., Banach algebra techniques in the operator theory, (1972), Academic Press New York-London · Zbl 0247.47001
[8] Garnett, J.B., Bounded analytic functions, (1981), Academic Press New York · Zbl 0469.30024
[9] Gorkin, P.; Zheng, D., Essentially commuting Toeplitz operators, Pacific J. math., 190, 87-109, (1999) · Zbl 1092.47504
[10] Gu, C., Products of several Toeplitz operators, J. funct. anal., 171, 483-527, (2000) · Zbl 0967.47021
[11] Gu, C.; Zheng, D., Products of block Toeplitz operators, Pacific J. math., 185, 115-148, (1998)
[12] K. Guo, D. Zheng, The distribution function inequality and block Toeplitz operators, preprint, 2001.
[13] Hoffman, K., Bounded analytic functions and Gleason parts, Ann. math., 86, 74-111, (1967) · Zbl 0192.48302
[14] Marshall, D.E., Subalgebras of L∞ containing H∞, Acta math., 137, 91-98, (1976) · Zbl 0334.46061
[15] Martinez-Avendaño, R., When do Toeplitz and Hankel operators commute?, Integral equations operator theory, 37, 341-349, (2000) · Zbl 0961.47015
[16] Nikolskii, N.K., Treatise on the shift operator, (1985), Springer New York, NY
[17] Power, S., Hankel operators on Hilbert space, (1982), Pittman Publishing Boston
[18] Sarason, D., Function theory on the unit circle, (1979), Virginia Polytechnic Institute and State University Blacksburg, VA
[19] D. Sarason, Holomorphic Spaces: A brief and selective survey, in: Holomorphic Spaces (Berkeley, CA, 1995), Mathematical Sciences Research Institute Publications, Vol. 33, Cambridge University Press, Cambridge, 1998, pp. 1-34. · Zbl 1128.47312
[20] Stroethoff, K.; Zheng, D., Products of Hankel and Toeplitz operator on the Bergman space, J. funct. anal., 169, 289-313, (1999) · Zbl 0945.47019
[21] Volberg, A., Two remarks concerning the theorem of S. axler, S.-Y.A. chang, and D. sarason, J. operator theory, 8, 209-218, (1982) · Zbl 0489.47015
[22] Younis, R.; Zheng, D., A distance formula and Bourgain algebras, Math. proc. Cambridge philos. soc., 120, 631-641, (1996) · Zbl 0878.46042
[23] Zheng, D., The distribution function inequality and products of Toeplitz operators and Hankel operators, J. funct. anal., 138, 477-501, (1996) · Zbl 0865.47019
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