Abstract
To overcome the unboundedness of the half-plane, we use Khinchine’s inequality and atom decomposition techniques to provide joint Carleson measure characterizations when the difference of composition operators is bounded or compact from standard weighted Bergman spaces to Lebesgue spaces over the half-plane for all index choices. For applications, we obtain direct analytic characterizations of the bounded and compact differences of composition operators on such spaces. This paper concludes with a joint Carleson measure characterization when the difference of composition operators is Hilbert-Schmidt.
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Cho H, Choe B, Koo H. Linear combinations of composition operators on the Fock-Sobolev spaces. Potential Anal, 2014, 41: 1223–1246
Choe B, Hosokawa T, Koo H. Hilbert-Schmidt differences of composition operators on the Bergman space. Math Z, 2011, 269: 751–775
Choe B, Izuchi K, Koo H. Linear sums of two composition operators on the Fock space. J Math Anal Appl, 2010, 369: 112–119
Choe B, Koo H, Park I. Compact differences of composition operators over polydisks. Integral Equations Operator Theory, 2012, 73: 57–91
Choe B, Koo H, Park I. Compact differences of composition operators on the Bergman spaces over the ball. Potential Anal, 2014, 40: 81–102
Choe B, Koo H, Smith W. Difference of composition operators over the half-plane. Trans Amer Math Soc, 2017, 369: 3173–3205
Choe B, Koo H, Wang M. Compact double differences of composition operators on the Bergman spaces. J Funct Anal, 2017, 272: 2273–2307
Choe B, Koo H, Wang M, et al. Compact linear combinations of composition operators induced by linear fractional maps. Math Z, 2015, 280: 807–824
Choe B, Koo H, Yi H. Positive Toeplitz operators between the harmonic Bergman spaces. Potential Anal, 2002, 17: 307–335
Choe B, Yi H. Representations and interpolations of harmonic Bergman functions on half-spaces. Nagoya Math J, 1998, 151: 51–89
Cowen C, Maccluer B. Composition Operators on Spaces of Analytic Functions. New York: CRC Press, 1995
Elliiott S, Wynn A. Composition operators on weighted Bergman spaces of a half-plane. Proc Edinb Math Soc (2), 2011, 54: 373–379
Halmos P. Measure Theory. New York: Springer-Verlag, 1974
Koo H, Wang M. Joint Carleson measure and the difference of composition operators on \(A_\alpha ^p\left({{B_n}} \right)\). J Math Anal Appl, 2014, 419: 1119–1142
Koo H, Wang M. Cancellation properties of composition operators on Bergman spaces. J Math Anal Appl, 2015, 432: 1174–1182
Kriete T, Moorhouse J. Linear relations in the Calkin algebra for composition operators. Trans Amer Math Soc, 2007, 359: 2915–2944
Luecking D. Embedding theorems for spaces of analytic functions via Khinchine’s inequality. Michigan Math J, 1993, 40: 333–358
Moorhouse J. Compact differences of composition operators. J Funct Anal, 2005, 219: 70–92
Saukko E. Difference of composition operators between standard weighted Bergman spaces. J Math Anal Appl, 2011, 381: 789–798
Saukko E. An application of atomic decomposition in Bergman spaces to the study of differences of composition operators. J Funct Anal, 2012, 262: 3872–3890
Shapiro J. Composition Operators and Classcical Function Theory. New York: Springer-Verlag, 1993
Shapiro J, Smith W. Hardy spaces that support no compact composition operators. J Funct Anal, 2003, 205: 62–89
Wang M, Pang C. Compact double differences of composition operators over the half-plane. Complex Anal Oper Theory, 2018, 12: 261–292
Zhu K. Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics. New York: Springer-Verlag, 2005
Zhu K. Operator Theory in Function Spaces, 2nd ed. Mathematical Surveys and Monographs. Providence: Amer Math Soc, 2007
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11771340 and 11431011). The authors thank the anonymous reviewers for their meaningful advice which improves the final version.
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Pang, C., Wang, M. Difference of composition operators over the half-plane. Sci. China Math. 63, 2299–2320 (2020). https://doi.org/10.1007/s11425-018-9439-2
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DOI: https://doi.org/10.1007/s11425-018-9439-2
Keywords
- weighted Bergman space
- joint Carleson measure
- composition operator
- Khinchine’s inequality
- atom decomposition
- Hilbert-Schmidt