Cancellation properties of composition operators on Bergman spaces. (English) Zbl 1321.47062
Summary: The compact difference of two composition operators on the Bergman spaces over the unit disc is characterized in in terms of certain cancellation property of the inducing maps at every “bad” boundary points, which make each single composition operator not be compact. In this paper, we completely characterize the compactness of a linear combination of three composition operators on the Bergman space. As one consequence of this characterization, we show that there is no cancellation property for the compactness of double difference of composition operators. More precisely, we show that if \(\varphi_i\) are distinct and none of \(C_{\varphi_i}\) is compact, then \((C_{\varphi_1} - C_{\varphi_2}) -(C_{\varphi_3} - C_{\varphi_1})\) is compact if and only if both \((C_{\varphi_1} - C_{\varphi_2})\) and \((C_{\varphi_3} - C_{\varphi_1})\) are compact.
References:
| [1] | Berkson, E., Composition operators isolated in the uniform operator topology, Proc. Amer. Math. Soc., 81, 230-232 (1981) · Zbl 0464.30027 |
| [2] | Bourdon, P., Components of linear fractional composition operators, J. Math. Anal. Appl., 279, 228-245 (2003) · Zbl 1043.47021 |
| [3] | Choe, B.; Koo, H.; Park, I., Compact differences of composition operators on the Bergman spaces over the ball, Potential Anal., 40, 81-102 (2014) · Zbl 1281.47012 |
| [4] | Choe, B.; Koo, H.; Wang, M.; Yang, J., Compact linear combinations of composition operators induced by linear fractional maps, Math. Z., 280, 807-824 (2015) · Zbl 1326.47022 |
| [5] | Cowen, C.; MacCluer, B., Composition Operators on Spaces of Analytic Functions (1995), CRC Press: CRC Press New York · Zbl 0873.47017 |
| [6] | Gallardo-Gutierrez, E.; Gonzalez, M.; Nieminen, P.; Saksman, E., On the connected component of compact composition operators on the Hardy space, Adv. Math., 219, 986-1001 (2008) · Zbl 1187.47021 |
| [7] | Koo, H.; Wang, M., Joint Carleson measure and the difference of composition operators on \(A_\alpha^p(B_n)\), J. Math. Anal. Appl., 419, 1119-1142 (2014) · Zbl 1294.47038 |
| [8] | Kriete, T.; Moorhouse, J., Linear relations in the Calkin algebra for composition operators, Trans. Amer. Math. Soc., 359, 2915-2944 (2007) · Zbl 1115.47023 |
| [9] | MacCluer, B., Components in the space of composition operators, Integral Equations Operator Theory, 12, 725-738 (1989) · Zbl 0685.47027 |
| [10] | MacCluer, B.; Shapiro, J., Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math., 38, 878-906 (1986) · Zbl 0608.30050 |
| [11] | Moorhouse, J., Compact differences of composition operators, J. Funct. Anal., 219, 70-92 (2005) · Zbl 1087.47032 |
| [12] | Moorhouse, J.; Toews, C., Differences of composition operators, (Trends in Banach Spaces and Operator Theory. Trends in Banach Spaces and Operator Theory, Contemp. Math., vol. 321 (2003)), 207-213 · Zbl 1052.47018 |
| [13] | Nieminen, P.; Saksman, E., On compactness of the difference of composition operators, J. Math. Anal. Appl., 298, 501-522 (2004) · Zbl 1072.47021 |
| [14] | Saukko, E., Difference of composition operators between standard weighted Bergman spaces, J. Math. Anal. Appl., 381, 789-798 (2011) · Zbl 1244.47024 |
| [15] | Saukko, E., An application of atomic decomposition in Bergman spaces to the study of differences of composition operators, J. Funct. Anal., 262, 3872-3890 (2012) · Zbl 1276.47032 |
| [16] | Shapiro, J., The essential norm of a composition operator, Ann. of Math., 125, 375-404 (1987) · Zbl 0642.47027 |
| [17] | Shapiro, J., Composition Operators and Classical Function Theory (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0791.30033 |
| [18] | Shapiro, J.; Sundberg, C., Isolation amongst the composition operators, Pacific J. Math., 145, 117-152 (1990) · Zbl 0732.30027 |
| [19] | Shapiro, J.; Taylor, P., Compact nuclear and Hilbert-Schmidt operators on \(H^2\), Indiana Univ. Math. J., 23, 471-496 (1973) · Zbl 0276.47037 |
| [20] | Zhu, K., Spaces of Holomorphic Functions in the Unit Ball (2005), Springer-Verlag: Springer-Verlag New York · Zbl 1067.32005 |
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