Finite rank product theorems for Toeplitz operators on the half-space. (English) Zbl 1173.47017

The authors study products of Toeplitz operators with harmonic symbols acting on the harmonic Bergman space over the half plane in \(\mathbb{R}^n\). The main result of the paper states that, if the product of a finite number of Toeplitz operators, whose harmonic symbols have certain boundary smoothness, has finite rank, then one of the symbols must be identically zero. Note that the number of factors is related to the dimension \(n\).


47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Full Text: DOI Link


[1] S. Axler, P. Bourdon and W. Ramey, Harmonic function theory, Springer-Verlag, New York, 1992. · Zbl 0765.31001
[2] P. Ahern and R. Schneider, Holomorphic Lipschitz functions in pseudoconvex domains, Amer. J. Math., 101 (1979), 543-565. · Zbl 0455.32008 · doi:10.2307/2373797
[3] B. R. Choe and H. Koo, Zero products of Toeplitz operators with harmonic symbols, J. Funct. Anal., 233 (2006), 307-334. · Zbl 1099.47028 · doi:10.1016/j.jfa.2005.08.007
[4] B. R. Choe, H. Koo and Y. J. Lee, Zero products of Toeplitz products with \(n\)-harmonic symbols, Integral Equations Operator Theory, 57 (2007), 43-66. · Zbl 1126.47028 · doi:10.1007/s00020-006-1444-2
[5] B. R. Choe, H. Koo and Y. J. Lee, Finite rank Toeplitz products with harmonic symbols, J. Math. Anal. Appl., 343 (2008), 81-98. · Zbl 1204.32003 · doi:10.1016/j.jmaa.2008.01.019
[6] B. Choe, H. Koo and H. Yi, Positive Toeplitz operators between the harmonic Bergman spaces, Potential Analysis, 17 (2002), 307-335. · Zbl 1014.47014 · doi:10.1023/A:1016356229211
[7] B. Choe and H. Yi, Representations and interpolations of harmonic Bergman functions on half spaces, Nagoya Math. J., 151 (1998), 51-89. · Zbl 0954.31004
[8] H. Kang and H. Koo, Estimates of the harmonic Bergman kernel on smooth domains, J. Funct. Anal., 185 (2001), 220-239. · Zbl 0983.31004 · doi:10.1006/jfan.2001.3761
[9] W. Ramey and H. Yi, Harmonic Bergman functions on half-spaces, Trans. Amer. Math. Soc., 348 (1996), 633-660. · Zbl 0848.31004 · doi:10.1090/S0002-9947-96-01383-9
[10] F. Ricci and M. Taibleson, Representation theorems for harmonic functions in mixed norm spaces on the half plane, Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), Rend. Circ. Mat. Palermo (2) suppl., 1981, pp. 121-127. · Zbl 0483.31001
[11] F. Ricci and M. Taibleson, Boundary values of harmonic functions in mixed norm spaces and their atomic structure, Ann. Scuola Norm. Sup. Pisa CI. Sci. (4), 10 (1983), 1-54. · Zbl 0527.30040
[12] W. Rudin, Function theory in the unit ball of \(\mathbf{C}^{n}\), Springer-Verlag, New York, 1980. · Zbl 0495.32001
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.