On Berezin type operators and Toeplitz operators on Bergman spaces. (English) Zbl 07717096
Summary: We introduce a class of integral operators called Berezin type operators. It is a generalization of the Berezin transform, and has a close relation to the Bergman-Carleson measures. The concept is partly motivated by the relationship between Hardy-Carleson measures and area operators. We mainly study the boundedness and the compactness of Berezin-type operators from a Bergman space \(A^{p_1}_{\alpha_1}\) to a Lebesgue space \(L^{p_2}_{\alpha_2}\) with \(0<p_1,p_2\leq \infty\) and \(\alpha_1,\alpha_2>-1\). We also show that Berezin-type operators are closely related to Toeplitz operators.
MSC:
| 47G10 | Integral operators |
| 47B38 | Linear operators on function spaces (general) |
| 47B34 | Kernel operators |
| 32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |
Keywords:
Berezin-type operators; Toeplitz operators; boundedness and compactness; Bergman-Carleson measuresReferences:
| [1] | Cohn, WS, Generalized area operators on Hardy spaces, J. Math. Anal. Appl., 216, 112-121 (1997) · Zbl 0903.46051 · doi:10.1006/jmaa.1997.5663 |
| [2] | Conway, JB, A Course in Functional Analysis (1985), New York: Springer-Verlag, New York · Zbl 0558.46001 |
| [3] | Gong, M.; Lou, Z.; Wu, Z., Area operators from \({\cal{H} }^p\) spaces to \(L^q\) spaces, Sci. China Math., 53, 357-366 (2010) · Zbl 1204.47042 · doi:10.1007/s11425-010-0031-9 |
| [4] | Liu, C., Sharp Forelli-Rudin estimates and the norm of the Bergman projection, J. Funct. Anal., 268, 255-277 (2015) · Zbl 1310.32007 · doi:10.1016/j.jfa.2014.09.027 |
| [5] | Luecking, D., Embedding derivatives of Hardy spaces into Lebesgue spaces, Proc. Lond. Math. Soc., 63, 595-619 (1991) · Zbl 0774.42011 · doi:10.1112/plms/s3-63.3.595 |
| [6] | Pau, J.; Zhao, R., Carleson measures and Toeplitz operators for weighted Bergman spaces on the unit ball, Michigan Math. J., 64, 759-796 (2015) · Zbl 1333.32007 · doi:10.1307/mmj/1447878031 |
| [7] | Rudin, W., Function Theory in the Unit Ball of \({\mathbb{C} }^n (1980)\), New York: Springer-Verlag, New York · Zbl 0495.32001 · doi:10.1007/978-1-4613-8098-6 |
| [8] | Videnskii, V.: On an analogue of Carleson measures. Dokl Akad Nauk SSSR 298, 1042-1047 (1988); Soviet Math Dokl. 37, 186-190 (1988) · Zbl 0687.30026 |
| [9] | Wu, Z., Area operator on Bergman spaces, Sci. China Ser. A, 49, 987-1008 (2006) · Zbl 1105.30019 · doi:10.1007/s11425-006-0987-7 |
| [10] | Zhao, R.; Zhou, L., \(L^p-L^q\) boundedness of Forelli-Rudin type operators on the unit ball of \({\mathbb{C} }^n\), J. Funct. Anal., 282 (2022) · Zbl 1539.47078 · doi:10.1016/j.jfa.2021.109345 |
| [11] | Zhao, R.; Zhu, K., Theory of Bergman spaces in the unit ball of \({\mathbb{C} }^n\), Mem. Soc. Math. Fr. (N.S.), 115, 103 (2008) · Zbl 1176.32001 |
| [12] | Zhu, K., Spaces of Holomorphic Functions in the Unit Ball (2005), New York: Springer-Verlag, New York · Zbl 1067.32005 |
| [13] | Zhu, K.: Operator Theory in Function Spaces. Math. Surveys Monogr, vol 138, 2nd edn. Amer. Math. Soc., Providence (2007) · Zbl 1123.47001 |
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.
