## Reducing subspaces of some multiplication operators on the Bergman space over polydisk.(English)Zbl 1356.47011

Summary: We consider the reducing subspaces of $$M_{z^N}$$ on $$A_\alpha^2(\mathbb{D}^k)$$, where $$k \geq 3$$, $$z^N = z_1^{N_1} \cdots z_k^{N_k}$$, and $$N_i \neq N_j$$ for $$i \neq j$$. We prove that each reducing subspace of $$M_{z^N}$$ is a direct sum of some minimal reducing subspaces. We also characterize the minimal reducing subspaces in the cases that $$\alpha = 0$$ and $$\alpha \in(- 1, + \infty) \smallsetminus \mathbb{Q}$$, respectively. Finally, we give a complete description of minimal reducing subspaces of $$M_{z^N}$$ on $$A_\alpha^2(\mathbb{D}^3)$$ with $$\alpha > - 1$$.

### MSC:

 47A15 Invariant subspaces of linear operators 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 46E20 Hilbert spaces of continuous, differentiable or analytic functions 32A36 Bergman spaces of functions in several complex variables

### Keywords:

minimal reducing subspaces
Full Text:

### References:

 [1] Guo, K.; Huang, H., Commutants, reducing subspaces and von Neumann algebras: based on multiplication operators on the Bergman space [2] Sun, S. L.; Wang, Y., Reducing subspaces of certain analytic Toeplitz operators on the Bergman space, Northeastern Mathematical Journal, 14, 2, 147-158, (1998) · Zbl 0923.47014 [3] Zhu, K., Reducing subspaces for a class of multiplication operators, Journal of the London Mathematical Society, 62, 2, 553-568, (2000) · Zbl 1158.47309 [4] Stessin, M.; Zhu, K., Reducing subspaces of the weighted shift operators, Proceedings of the American Mathematical Society, 130, 9, 2631-2639, (2002) · Zbl 1035.47015 [5] Hu, J.; Sun, S.; Xu, X.; Yu, D., Reducing subspace of analytic Toeplitz operators on the Bergman space, Integral Equations and Operator Theory, 49, 3, 387-395, (2004) · Zbl 1077.47030 [6] Xu, A.; Yan, C., Reducing subspace of analytic Toeplitz operators on weighted Bergman spaces, Chinese Annals of Mathematics, Series A, 30, 639-646, (2009) · Zbl 1211.47058 [7] Guo, K.; Sun, S.; Zheng, D.; Zhong, C., Multiplication operators on the Bergman space via the Hardy space of the bidisk, Journal für Die Reine und Angewandte Mathematik, 2009, 628, 129-168, (2009) · Zbl 1216.47055 [8] Douglas, R. G.; Sun, S.; Zheng, D., Multiplication operators on the Bergman space via analytic continuation, Advances in Mathematics, 226, 1, 541-583, (2011) · Zbl 1216.47053 [9] Guo, K.; Huang, H., On multiplication operators of the Bergman space: similarity, unitary equivalence and reducing subspaces, Journal of Operator Theory, 65, 2, 355-378, (2011) · Zbl 1222.47040 [10] Sun, S.; Zheng, D.; Zhong, C., Classification of reducing subspaces of a class of multiplication operators on the Bergman space via the hardy space of the bidisk, Canadian Journal of Mathematics, 62, 2, 415-438, (2010) · Zbl 1185.47030 [11] Douglas, R. G.; Putinar, M.; Wang, K., Reducing subspaces for analytic multipliers of the Bergman space, Journal of Functional Analysis, 263, 6, 1744-1765, (2012) · Zbl 1275.47071 [12] Guo, K.; Huang, H., Multiplication operators defined by covering maps on the Bergman space: the connection between operator theory and von Neumann algebras, Journal of Functional Analysis, 260, 4, 1219-1255, (2011) · Zbl 1216.46054 [13] Guo, K.; Huang, H., Geometric constructions of thin Blaschke products and reducing subspace problem, Proceedings of the London Mathematical Society, 109, 4, 1050-1091, (2014) · Zbl 1305.47026 [14] Lu, Y.; Zhou, X., Invariant subspaces and reducing subspaces of weighted Bergman space over bidisk, Journal of the Mathematical Society of Japan, 62, 3, 745-765, (2010) · Zbl 1202.47008 [15] Shan, L., Reducing subspaces for a class of analytic Toeplitz operators on bidisc, Journal of Fudan University, 42, 2, 196-200, (2003) [16] Shi, Y.; Lu, Y., Reducing subspaces for Toeplitz operators on the polydisk, Bulletin of the Korean Mathematical Society, 50, 2, 687-696, (2013) · Zbl 1280.47039 [17] Zhou, X.; Shi, Y.; Lu, Y., Invariant subspaces and reducing subspaces of weighted Bergman space over polydisc, Scientia Sinica Mathematica, 41, 5, 427-438, (2011) [18] Dan, H.; Huang, H., Multiplication operators defined by a class of polynomials on $$L_\alpha^2$$($$D^2$$), Integral Equations and Operator Theory, 80, 4, 581-601, (2014) · Zbl 1302.47061
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.