The boundedness and compactness of weighted composition operators have been extensively investigated on various analytic function spaces. In this paper, we study the boundedness and compactness of two several variables differences weighted composition operators on some analytic function spaces.
Citation: Aydah Mohammed Ayed Al-Ahmadi. Differences weighted composition operators in several variables between some spaces of analytic functions[J]. AIMS Mathematics, 2023, 8(11): 27363-27375. doi: 10.3934/math.20231400
The boundedness and compactness of weighted composition operators have been extensively investigated on various analytic function spaces. In this paper, we study the boundedness and compactness of two several variables differences weighted composition operators on some analytic function spaces.
[1] | R. F. Allen, F. Colonna, Weighted composition operators from ${\bf{H}}_\alpha$ to the Bloch space of a bounded homogeneous domain, Integr. Equat. Oper. Th., 66 (2010), 21–40. https://doi.org/10.1007/s00020-009-1736-4 doi: 10.1007/s00020-009-1736-4 |
[2] | J. Bonet, M. Lindstrom, E. Wolf, Differences of composition operators between weighted Banach spaces of holomorphic functions, J. Aust. Math. Soc., 84 (2008), 9–20. https://doi.org/10.1017/S144678870800013X doi: 10.1017/S144678870800013X |
[3] | B. S. Bourdon, J. A. Cima, A. L. Matheson, Compact composition operators on BMOA, T. Am. Math. Soc. 351 (1999), 2183–2169. |
[4] | B. Choe, H. Koo, I. Park, Compact differences of composition operators over polydisks, Integr. Equat. Oper. Th., 73 (2013), 57–91. https://doi.org/10.1007/s00020-012-1962-z doi: 10.1007/s00020-012-1962-z |
[5] | C. Cowen, B. Maccluer, Composition operators on spaces of analytic functions, Boca Raton: CRC Press, 1995. |
[6] | X. Fu, Differences of weighetd composition operator from weighted Bergman spaces to weighted-type spaces, Bull. Math. Anal. Appl., 5 (2013), 65–70. |
[7] | A. E. S. Ahmed, M. A. Bakhit, Holomorphic ${{\mathcal N}_K}$ and Bergman-type spaces, Birkhuser Series on Operator Theory: Advances and Applications, Basel: BirkhuserVerlag Publisher, 195 (2009), 121–138. https://doi.org/10.1007/978-3-0346-0174-0-5 |
[8] | A. E. S. Ahmed, M. A. Bakhit, Hadamard products and ${\mathcal N}_K$ spaces, Math. Comput. Model., 51 (2010), 33–43. https://doi.org/10.1016/j.mcm.2009.08.037 doi: 10.1016/j.mcm.2009.08.037 |
[9] | K. Heller, B. D. Maccluer, R. J. Weir, Compact differences of composition operators in several variables, Integr. Equat. Oper. Th., 69 (2011), 419–428. https://doi.org/10.1007/s00020-010-1840-5 doi: 10.1007/s00020-010-1840-5 |
[10] | T. Hosokawa, S. Ohno, Differences of weighted composition operators from $H^\infty$ to Bloch space, Taiwan. J. Math., 16 (2012), 2093–2105. https://doi.org/10.11650/twjm/1500406842 doi: 10.11650/twjm/1500406842 |
[11] | T. Hosokawa, Differences of weighted composition operators on the Bloch spaces, Complex Anal. Oper. Th., 3 (2009), 847–866. https://doi.org/10.1007/s11785-008-0062-1 doi: 10.1007/s11785-008-0062-1 |
[12] | T. Hosokawa, S. Ohno, Differences of composition operators on the Bloch spaces, J. Operat. Theor., 57 (2007), 229–242. |
[13] | L. Jiang, C. Ouyang, Compact differences of composition operators on holomorphic function spaces in the unit ball, Acta Math. Sci., 31 (2011), 1679–1693. https://doi.org/10.1016/S0252-9602(11)60291-9 doi: 10.1016/S0252-9602(11)60291-9 |
[14] | B. Hu, L. H. Khoi, Compact difference of weighted composition operators on ${\mathcal N}_p$ spaces in the ball, Roum. J. Pure Appl. Math., 60 (2015), 101–116. |
[15] | A. E. Shammahy, Weighted composition operators acting between kind of weighted Bergman-type spaces and the Bers-type space, Inter. J. Math. Comput. Phys. Electr. Comput. Eng., 8 (2014), 496–499. |
[16] | M. Lindstrom, E. Wolf, Essential norm of the difference of weighted composition operators, Monatsh. Math., 153 (2000), 133–143. https://doi.org/10.1007/s00605-007-0493-1 doi: 10.1007/s00605-007-0493-1 |
[17] | P. J. Nieminen, Compact differences of composition operators on Bloch and Lipschitz spaces, Comput. Meth. Funct. Th., 7 (2007), 325–344. https://doi.org/10.1007/BF03321648 doi: 10.1007/BF03321648 |
[18] | C. Shaolin, H. Hidetaka, Z. J. Feng, Composition operators on Bloch and Hardy type spaces, Math. Z., 301 (2022), 3939–3957. https://doi.org/10.1007/s00209-022-03046-z doi: 10.1007/s00209-022-03046-z |
[19] | C. Shaolin, H. Hidetaka, Harmonic Lipschitz type spaces and composition operators meet majorants, J. Geom. Anal., 33 (2023), 181. https://doi.org/10.1007/s12220-023-01232-x doi: 10.1007/s12220-023-01232-x |
[20] | J. H. Shapiro, Composition operators and classical function theory, New York: Springer-Verlag, 1993. https://doi.org/10.1007/978-1-4612-0887-7 |
[21] | R. Zhao, On $\alpha$-Bloch functions and VMOA, Acta Math. Sci., 3 (1996), 349–360. https://doi.org/10.1016/S0252-9602(17)30811-1 doi: 10.1016/S0252-9602(17)30811-1 |
[22] | K. Zhu, Operator theory in function spaces, New York: Marcel Dekker, 2007. https://doi.org/10.1090/surv/138 |
[23] | X. Zhua, W. Yang, Differences of composition operators from weighted Bergman spaces to Bloch spaces, Filomat, 28 (2014), 1935–1941. https://doi.org/10.2298/FIL1409935Z doi: 10.2298/FIL1409935Z |
[24] | S. Ueki, Weighted composition operators acting between the ${\mathcal N}_p$-space and the Weighted-type space ${\mathcal H}_\alpha^\infty$, Indagat. Math., 23 (2012), 243–255. https://doi.org/10.1016/j.indag.2011.11.006 doi: 10.1016/j.indag.2011.11.006 |
[25] | E. Wolf, Weighted composition operators between weighted Bloch type spaces, B. Soc. Royale Sci. Liège, 80 (2011), 806–816. |