×

zbMATH — the first resource for mathematics

Compact double differences of composition operators on the Bergman spaces over the ball. (English) Zbl 1437.47011
Summary: B. R. Choe et al. [J. Funct. Anal. 272, No. 6, 2273–2307 (2017; Zbl 1437.47010)] have recently characterized compact double differences formed by four composition operators acting on the standard weighted Bergman spaces over the disk of the complex plane. In this paper, we extend such a result to the ball setting. Our characterization is obtained under a suitable restriction on inducing maps, which is automatically satisfied in the case of the disk. We exhibit concrete examples, for the first time even for single composition operators, which shows that such a restriction is essential in the case of the ball.
MSC:
47B33 Linear composition operators
30H20 Bergman spaces and Fock spaces
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] B. R. Choe, H. Koo and I. Park, Compact differences of composition operators over polydisks, Integral Equations Operator Theory 73(1) (2012), 57-91. · Zbl 1339.47028
[2] B. R. Choe, H. Koo and I. Park, Compact differences of composition operators on the Bergman spaces over the ball, Potential Analysis 40 (2014), 81-102. · Zbl 1281.47012
[3] B. R. Choe, H. Koo and M. Wang, Compact double differences of composition operators on the Bergman spaces, J. Funct. Anal. 272 (2017), 2273-2307. · Zbl 1437.47010
[4] D. Clahane, Compact composition operators on weighted Bergman spaces of the unit ball, J. Oper. Theory 45 (2001), 335-355. · Zbl 0999.47023
[5] C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic fuctions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. · Zbl 0873.47017
[6] P. Halmos, Measure Theory, Springer-Verlag, New York, 1974. · Zbl 0283.28001
[7] H. Koo and W. Smith, Composition operators induced by smooth self-maps of the unit ball in \(\mathbf{C}^N\), J. Math. Anal. Appl. 329 (2007), 617-633. · Zbl 1115.32004
[8] H. Koo and M. Wang, Joint Carleson measure and the difference of composition operators on \(A^p_\alpha(\mathbf{B}_n)\), J. Math. Anal. Appl. 419 (2014), 1119-1142. · Zbl 1294.47038
[9] J. Moorhouse, Compact differences of composition operators, J. Funct. Anal. 219 (2005), 70-92. · Zbl 1087.47032
[10] W. Rudin, Function theory in the unit ball of \(\mathbf{C}^n\), Springer, New York (1980). · Zbl 0495.32001
[11] J. H. Shapiro, Composition operators and classical function theory, Springer, New York, 1993. · Zbl 0791.30033
[12] K. Zhu, Spaces of holomorphic functions in the unit ball, Springer-Verlag, New York, 2005. · Zbl 1067.32005
[13] K. Zhu, Compact composition operators on Bergman spaces of the unit ball, Houston J. Math. 33 (2007), 273-283. · Zbl 1114.47031
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.