Reducibility for a class of analytic multipliers on Sobolev disk algebra. (English) Zbl 1502.47044
Summary: We prove the reducibility of analytic multipliers \(M_\phi\) with a class of finite Blaschke products symbol \(\phi\) on the Sobolev disk algebra \(R(\mathbb{D})\). We also describe their nontrivial minimal reducing subspaces.
MSC:
| 47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
| 46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |
| 30J10 | Blaschke products |
References:
| [1] | Adams, R. A., Sobolev Spaces (1975), New York: Academic Press, New York · Zbl 0314.46030 |
| [2] | Cowen, C., The commutant of an analytic Toeplitz oprator, Trans Amer Math Soc, 239, 1-31 (1978) · Zbl 0391.47014 · doi:10.1090/S0002-9947-1978-0482347-9 |
| [3] | Chen, Y.; Xu, H. M., Reducibility and unitarily equivalence for a class of analytic multipliers on the Dirichlet space, Complex Anal Oper Theory, 7, 6, 1897-1908 (2013) · Zbl 1291.47021 · doi:10.1007/s11785-012-0267-1 |
| [4] | Chen, Y.; Lee, Y. J.; Yu, T., Reducibility and unitary equivalence for a class of analytic multiplication operator on the Dirichlet space, Studia Math, 220, 2, 141-156 (2014) · Zbl 1298.47047 · doi:10.4064/sm220-2-3 |
| [5] | Chen, Y.; Qin, C. T.; Wu, Q., Reducibility and unitary equivalence of analytic multipliers on Sobolev disk algebra, J Math Anal Appl, 455, 1249-1256 (2017) · Zbl 1372.30060 · doi:10.1016/j.jmaa.2017.06.030 |
| [6] | Chen, Y.; Xu, X. M.; Zhao, Y. L., Reducibility for a class of analytic multipliers on the Dirichlet space, Complex Anal Oper Theory, 12, 1781-1790 (2018) · Zbl 06984206 · doi:10.1007/s11785-017-0661-9 |
| [7] | Douglas, R.; Putinar, M.; Wang, K., Reducing subspaces for analytic multipliers of the Bergman space, J Funct Anal, 263, 6, 1744-1765 (2012) · Zbl 1275.47071 · doi:10.1016/j.jfa.2012.06.008 |
| [8] | Gu, C. X.; Luo, S. B.; Xiao, J., Reducing subspaces for multiplication operators on the Dirichlet space through local inverses and Riemann surfaces, Complex Manifolds, 4, 84-119 (2017) · Zbl 06815228 · doi:10.1515/coma-2017-0007 |
| [9] | Guo, K. Y.; Sun, S. H.; Zheng, D. C.; Zhong, C. Y., Multiplication operators on the Bergman space via the Hardy space of the bidisk, J Reine Angew Math, 628, 129-168 (2009) · Zbl 1216.47055 |
| [10] | Hu, J. Y.; Sun, S. H.; Xu, X. M.; Yu, D. H., Reducing subspace of analytic Toeplitz operators on the Bergman space, Integr Equ Oper Theory, 49, 387-395 (2004) · Zbl 1077.47030 · doi:10.1007/s00020-002-1207-7 |
| [11] | Jiang, C. L.; Wang, Z. Y., Structure of Hilbert space operators (2006), Singapore: World Scientic, Singapore · Zbl 1102.47001 · doi:10.1142/5993 |
| [12] | Li, Y. C.; Liu, Q. J.; Lan, W. H., On similarity and reducing subspaces of multiplication operator on Sobolev disk algebra, J Math Anal Appl, 419, 2, 1161-1167 (2014) · Zbl 1311.47041 · doi:10.1016/j.jmaa.2014.05.052 |
| [13] | Luo, S. B., Reducing subspaces of multiplication operators on the Dirichlet space, Integr Equ Oper Theory, 85, 4, 539-554 (2016) · Zbl 06646409 · doi:10.1007/s00020-016-2295-0 |
| [14] | Stessin, M.; Zhu, K. H., Reducing subspaces of weighted shift operators, Proc Amer Math Soc, 130, 2631-2639 (2002) · Zbl 1035.47015 · doi:10.1090/S0002-9939-02-06382-7 |
| [15] | Stessin, M.; Zhu, K. H., Generalized factorization in Hardy spaces and the commutant of Toeplitz operators, Canad J Math, 55, 2, 379-400 (2003) · Zbl 1048.30021 · doi:10.4153/CJM-2003-017-1 |
| [16] | Sun, S. H.; Zheng, D. C.; Zhong, C. Y., Classification of reducing subspaces of a class of multiplication operators on the Bergman space via the Hardy space of the bidisk, Canad J Math, 62, 2, 415-438 (2010) · Zbl 1185.47030 · doi:10.4153/CJM-2010-026-4 |
| [17] | Thomson, J., The commutant of a class of analytic Toeplitz operators II, Indiana Univ Math J, 25, 793-800 (1976) · Zbl 0334.47023 · doi:10.1512/iumj.1976.25.25063 |
| [18] | Thomson, J., The commutant of a class of analytic Toeplitz operators, Amer J Math, 99, 522-529 (1977) · Zbl 0372.47018 · doi:10.2307/2373929 |
| [19] | Zhao, L. K., Reducing subspaces for a class of multiplication operators on the Dirichlet space, Proc Amer Math Soc, 137, 3091-3097 (2019) · Zbl 1182.47033 · doi:10.1090/S0002-9939-09-09859-1 |
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