×

Compact differences of weighted composition operators on the weighted Bergman spaces. (English) Zbl 1355.47013

Summary: In this paper, we consider the compact differences of weighted composition operators on the standard weighted Bergman spaces. Some necessary and sufficient conditions for the differences of weighted composition operators to be compact are given, which extends J. Moorhouse’s results in [J. Funct. Anal. 219, No. 1, 70–92 (2005; Zbl 1087.47032)].

MSC:

47B33 Linear composition operators
30H10 Hardy spaces
46E15 Banach spaces of continuous, differentiable or analytic functions

Citations:

Zbl 1087.47032
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Moorhouse, J: Compact differences of composition operators. J. Funct. Anal. 219, 70-92 (2005) · Zbl 1087.47032 · doi:10.1016/j.jfa.2004.01.012
[2] Contreras, M, Hernández-Díaz, A: Weighted composition operators on Hardy spaces. J. Math. Anal. Appl. 263, 224-233 (2011) · Zbl 1026.47016 · doi:10.1006/jmaa.2001.7610
[3] Lindström, M, Wolf, E: Essential norm of the difference of weighted composition operators. Monatshefte Math. 153, 133-143 (2008) · Zbl 1146.47015 · doi:10.1007/s00605-007-0493-1
[4] Mirzakarami, G, Seddighi, K: Weighted composition operators on Bergman and Dirichlet spaces. Georgian Math. J. 4, 373-383 (1997) · Zbl 0891.47018 · doi:10.1023/A:1022946629849
[5] Stević, S: Norms of some operators on the Bergman and the Hardy space in the unit polydisk and the unit ball. Appl. Math. Comput. 215, 2199-2205 (2009) · Zbl 1186.32004
[6] Stević, S, Ueki, S: Weighted composition operators from the weighted Bergman space to the weighted Hardy space on the unit ball. Appl. Math. Comput. 215, 3526-3533 (2010) · Zbl 1197.47040
[7] Ueki, S, Luo, L: Essential norms of weighted composition operators between weighted Bergman spaces of the ball. Acta Sci. Math. 74, 829-843 (2008) · Zbl 1199.30274
[8] Shapiro, J, Taylor, P: Compact nuclear and Hilbert Schmidt operators on \(H2H^2\). Indiana Univ. Math. J. 23, 471-496 (1973) · Zbl 0276.47037 · doi:10.1512/iumj.1974.23.23041
[9] MacCluer, B, Shapiro, J: Angular derivatives and compact composition operators on the Hardy and Bergman spaces. Can. J. Math. 38, 878-906 (1986) · Zbl 0608.30050 · doi:10.4153/CJM-1986-043-4
[10] Shapiro, J: The essential norm of a composition operator. Ann. Math. 125, 375-404 (1987) · Zbl 0642.47027 · doi:10.2307/1971314
[11] Berkson, E: Composition operators isolated in the uniform operator topology. Proc. Am. Math. Soc. 81, 230-232 (1981) · Zbl 0464.30027 · doi:10.1090/S0002-9939-1981-0593463-0
[12] Shapiro, J, Sundberg, C: Isolation amongst the composition operators. Pac. J. Math. 145, 117-152 (1990) · Zbl 0732.30027 · doi:10.2140/pjm.1990.145.117
[13] MacCluer, B: Components in the space of composition operators. Integral Equ. Oper. Theory 12, 725-738 (1989) · Zbl 0685.47027 · doi:10.1007/BF01194560
[14] 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 · doi:10.1016/j.aim.2008.06.005
[15] Allen, R, Heller, K, Pons, M: Compact differences of composition operators on weighted Dirichlet spaces. Cent. Eur. J. Math. 12, 1040-1051 (2014) · Zbl 1312.47027
[16] Jiang, Z, Stević, S: Compact differences of weighted composition operators from weighted Bergman spaces to weighted-type spaces. Appl. Math. Comput. 217, 3522-3530 (2010) · Zbl 1204.30043
[17] Michalska, M, Michalski, A: Necessary condition for compactness of a difference of composition operators on the Dirichlet space. J. Math. Anal. Appl. 426, 864-871 (2015) · Zbl 1308.47027 · doi:10.1016/j.jmaa.2015.02.002
[18] Stević, S: Essential norm of differences of weighted composition operators between weighted-type spaces on the unit ball. Appl. Math. Comput. 217, 1811-1824 (2010) · Zbl 1221.47062
[19] Stević, S, Jiang, Z: Compactness of the differences of weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball. Taiwan. J. Math. 15, 2647-2665 (2011) · Zbl 1315.47034
[20] Wolf, E: Compact differences of weighted composition operators between weighted Bergman spaces of infinite order. Houst. J. Math. 37, 1203-1209 (2011) · Zbl 1238.47018
[21] Wolf, E: Weakly compact differences of (weighted) composition operators. Asian-Eur. J. Math. 4, 695-703 (2011) · Zbl 1263.47030 · doi:10.1142/S1793557111000575
[22] Choe, B, Hosokawa, T, Koo, H: Hilbert-Schmidt differences of composition operators on the Bergman spaces. Math. Z. 269, 751-775 (2011) · Zbl 1234.47009 · doi:10.1007/s00209-010-0757-7
[23] 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 · doi:10.1007/s00209-015-1449-0
[24] Koo, H, Wang, M: Cancellation properties of composition operators on Bergman spaces. J. Math. Anal. Appl. 432, 1174-1182 (2015) · Zbl 1321.47062 · doi:10.1016/j.jmaa.2015.07.027
[25] Al-Rawashdeh, W, Narayan, S: Difference of composition operators on Hardy space. J. Math. Inequal. 7, 427-444 (2013) · Zbl 1294.47048 · doi:10.7153/jmi-07-38
[26] Cowen, C, MacCluer, B: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995) · Zbl 0873.47017
[27] Kriete, T, Moorhouse, J: Linear relations in the Calkin algebra for composition operators. Trans. Am. Math. Soc. 359, 2915-2944 (2007) · Zbl 1115.47023 · doi:10.1090/S0002-9947-07-04166-9
[28] Choe, B, Koo, H, Park, I: Compact differences of composition operators on the Bergman spaces over the ball. Potential Anal. 40, 81-92 (2014) · Zbl 1281.47012 · doi:10.1007/s11118-013-9343-z
[29] Zhu, K: Operator Theory in Function Spaces. Dekker, New York (1990) · Zbl 0706.47019
[30] Zhu, K: Compact composition operators on Bergman spaces of the unit ball. Houst. J. Math. 33, 273-283 (2007) · 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. 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.