Previous |  Up |  Next

Article

Keywords:
weighted composition operator; Hardy space; weighted Bergman space; essential norm; compact; difference
Summary:
In this paper, we limit our analysis to the difference of the weighted composition operators acting from the Hardy space to weighted-type space in the unit ball of $\mathbb {C}^N$, and give some necessary and sufficient conditions for their boundedness or compactness. The results generalize the corresponding results on the single weighted composition operators and on the differences of composition operators, for example, M. Lindström and E. Wolf: Essential norm of the difference of weighted composition operators. Monatsh. Math. 153 (2008), 133--143.
References:
[1] Allen, R. F., Colonna, F.: Weighted composition operators from $H^{\infty}$ to the Bloch space of a bounded homogeneous domain. Integral Equations Oper. Theory 66 (2010), 21-40. DOI 10.1007/s00020-009-1736-4 | MR 2591634
[2] Bonet, J., Domański, P., Lindström, M.: Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Can. Math. Bull. 42 (1999), 139-148. DOI 10.4153/CMB-1999-016-x | MR 1692002 | Zbl 0939.47020
[3] Bonet, J., Domański, P., Lindström, M., Taskinen, J.: Composition operators between weighted Banach spaces of analytic functions. J. Aust. Math. Soc., Ser. A 64 (1998), 101-118. DOI 10.1017/S1446788700001336 | MR 1490150 | Zbl 0912.47014
[4] Bonet, J., Lindström, M., Wolf, E.: Differences of composition operators between weighted Banach spaces of holomorphic functions. J. Aust. Math. Soc. 84 (2008), 9-20. DOI 10.1017/S144678870800013X | MR 2469264
[5] Cowen, C. C., MacCluer, B. D.: Composition Operators on Spaces of Analytic Functions. Studies in Advanced Mathematics Boca Raton, FL, CRC Press (1995). MR 1397026 | Zbl 0873.47017
[6] Dai, J. N., Ouyang, C. H.: Differences of weighted composition operators on $H_{\alpha}^\infty(B_N)$. J. Inequal. Appl. Article ID 127431 (2009), 19 pp. MR 2579553
[7] Fang, Z. S., Zhou, Z. H.: Differences of composition operators on the space of bounded analytic functions in the polydisc. Abstr. Appl. Anal. Article ID 983132 (2008), 10 pp. MR 2466222 | Zbl 1160.32009
[8] Fang, Z. S., Zhou, Z. H.: Differences of composition operators on the Bloch space in the polydisc. Bull. Aust. Math. Soc. 79 (2009), 465-471. DOI 10.1017/S0004972709000045 | MR 2505351 | Zbl 1166.47032
[9] Gorkin, P., Mortini, R., Suarez, D.: Homotopic composition operators on $H^\infty(B^n)$. Jarosz, Krzysztof (ed.), Function spaces. Proceedings of the 4th conference, Edwardsville, IL, USA (2002), Providence, RI: American Mathematical Society (AMS), Contemp. Math 328 177-188 (2003). MR 1990399
[10] Hosokawa, T., Izuchi, K., Ohno, S.: Topological structure of the space of weighted composition operators on $H^\infty$. Integral Equations Oper. Theory 53 (2005), 509-526. MR 2187435
[11] Hosokawa, T., Ohno, S.: Topological structures of the sets of composition operators on the Bloch spaces. J. Math. Anal. Appl. 314 (2006), 736-748. DOI 10.1016/j.jmaa.2005.04.080 | MR 2185263 | Zbl 1087.47029
[12] Hosokawa, T., Ohno, S.: Differences of composition operators on the Bloch spaces. J. Oper. Theory 57 (2007), 229-242. MR 2328996 | Zbl 1174.47019
[13] Lindström, M., Wolf, E.: Essential norm of the difference of weighted composition operators. Monatsh. Math. 153 (2008), 133-143. DOI 10.1007/s00605-007-0493-1 | MR 2373366
[14] MacCluer, B. D.: Compact composition operators on $H^p(B_N)$. Mich. Math. J. 32 (1985), 237-248. DOI 10.1307/mmj/1029003191 | MR 0783578 | Zbl 0585.47022
[15] MacCluer, B. D., Ohno, S., Zhao, R.: Topological structure of the space of composition operators on $H^\infty$. Integral Equations Oper. Theory 40 (2001), 481-494. MR 1839472
[16] Moorhouse, J.: Compact differences of composition operators. J. Funct. Anal. 219 (2005), 70-92. DOI 10.1016/j.jfa.2004.01.012 | MR 2108359 | Zbl 1087.47032
[17] Ohno, S., Stroethoff, K., Zhao, R.: Weighted composition operators between Bloch-type spaces. Rocky Mt. J. Math. 33 (2003), 191-215. DOI 10.1216/rmjm/1181069993 | MR 1994487 | Zbl 1042.47018
[18] Shapiro, J. H.: Composition Operators and Classical Function Theory. Universitext: Tracts in Mathematics New York, Springer (1993). MR 1237406 | Zbl 0791.30033
[19] Stević, S., Wolf, E.: Differences of composition operators between weighted-type spaces of holomorphic functions on the unit ball of $C^n$. Appl. Math. Comput. 215 (2009), 1752-1760. DOI 10.1016/j.amc.2009.07.036 | MR 2557418
[20] Toews, C.: Topological components of the set of composition operators on $H^{\infty}(B_N)$. Integral Equations Oper. Theory 48 (2004), 265-280. DOI 10.1007/s00020-002-1180-1 | MR 2030531
[21] Wolf, E.: Differences of composition operators between weighted Banach spaces of holomorphic functions on the unit polydisk. Result. Math. 51 (2008), 361-372. DOI 10.1007/s00025-007-0283-z | MR 2400173 | Zbl 1154.47018
[22] Yang, K. B., Zhou, Z. H.: Essential norm of the difference of composition operators on Bloch space. Czech. Math. J. 60 (2010), 1139-1152. DOI 10.1007/s10587-010-0079-2 | MR 2738975 | Zbl 1220.47045
[23] Zeng, H. G., Zhou, Z. H.: An estimate of the essential norm of a composition operator from $ F(p, q, s)$ to $\mathcal{B}^\alpha$ in the unit ball. J. Inequal. Appl. Article ID 132970 (2010), 22 pp. MR 2645968
[24] Zhou, Z. H., Chen, R. Y.: Weighted composition operators from $F(p, q, s)$ to Bloch type spaces. Int. J. Math. 19 (2008), 899-926. DOI 10.1142/S0129167X08004984 | MR 2446507 | Zbl 1163.47021
[25] Zhou, Z. H., Shi, J. H.: Compactness of composition operators on the Bloch space in classical bounded symmetric domains. Mich. Math. J. 50 (2002), 381-405. DOI 10.1307/mmj/1028575740 | MR 1914071 | Zbl 1044.47021
[26] Zhu, K. H.: Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics 226 Springer, New York (2005). MR 2115155 | Zbl 1067.32005
Partner of
EuDML logo