×

Toeplitz operators on harmonic Bergman spaces. (English) Zbl 0902.47026

Summary: We study Toeplitz operators on the harmonic Bergman space \(b^p(B)\), where \(B\) is the open unit ball in \(\mathbb{R}^n\) \((n\geq 2)\), for \(1< p<\infty\). We give characterizations for the Toeplitz operators with positive symbols to be bounded, compact, and in Schatten classes. We also obtain a compactness criteria for the Toeplitz operators with continuous symbols.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] J. Arazy, S. Fisher, and J. Peetre,Hankel operators on weighted Bergman spaces, Amer. J. Math. 110 (1988), 989-1054. · Zbl 0669.47017
[2] S. Axler,Bergman spaces and their operators, Surveys of Some Results in Operator Theory, vol. 1, (J.B. Conway and B.B. Morrel, editors), Pitman Res. Notes Math. Ser. vol. 171, 1988, pp. 1-50.
[3] S. Axler, P. Bourdon, and W. Ramey,Harmonic Function Theory, Springer-Verlag, New York, 1992. · Zbl 0765.31001
[4] R.R. Coifman, R. Rochberg,Representation theorems for holomorphic and harmonic functions, Ast?risque 77 (1980), 11-65. · Zbl 0472.46040
[5] M. Jovovi?,Compact Hankel operators on harmonic Bergman spaces, Integral Equations and Operator Theory 22 (1995), 295-304. · Zbl 0837.47021
[6] E. Ligocka,On the reproducing kernel for harmonic functions and the space of Bloch harmonic functions on the unit ball in R n , Studia Math. 87 (1987), 23-32. · Zbl 0658.31006
[7] E. Ligocka, Estimates in Sobolev norms ??? p 3 for harmonic and holomorphic fuctions and interpolation between Sobolev and Holder spaces of harmonic functions, Studia Math. 86 (1987), 255-271. · Zbl 0642.46035
[8] D.H. Luecking,Trace ideal criteria for Toeplity operators, J. Funct. Anal. 73 (1987), 345-368. · Zbl 0618.47018
[9] D.H. Luecking,Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disc, J. Funct. Anal. 110 (1992), 247-271. · Zbl 0773.47014
[10] O.L. Oleinik,Embedding theorems for weighted classes of harmonic and analytic functions, J. Soviet Math. 9 (1978), 228-243. · Zbl 0396.31001
[11] V.L. Oleinik, B.S. Pavlov,Embedding theorems for weighted classes of harmonic and analytic functions, J. Soviet Math. 2 (1974), 135-142. · Zbl 0278.46032
[12] E.M. Stein,Singular intergrals and differentiability properties of functions, Princeton University Press, Princeton, NJ, 1970. · Zbl 0207.13501
[13] Z. Wu,Operators on harmonic Bergman spaces, Preprint. · Zbl 0860.47017
[14] K. Zhu,Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains, J. Operator Theory 20 (1988), 329-357. · Zbl 0676.47016
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.