Kuwahara, Shuhei Reducing subspaces of weighted Hardy spaces on polydisks. (English) Zbl 1331.47050 Nihonkai Math. J. 25, No. 2, 77-83 (2014). Let \(H_w^2(\mathbb{D}^n)\) be the weighted Hardy space on the \(n\)-dimensional polydisk. Let \(M_{z_k}^{M_k}\) be the shift operator \(f(z) \mapsto f(z)z_k^{N_k}\) acting on \(H_w^2(\mathbb{D}^n)\).The article determines common reducing subspaces of the operators \(M_{z_1}^{M_1}, M_{z_2}^{M_2}, \dots, M_{z_n}^{M_n}\), where the numbers \(N_k\) are arbitrary nonnegative integers. The work is a generalization of a result in [M. Stessin and K. Zhu, Proc. Am. Math. Soc. 130, No. 9, 2631–2639 (2002; Zbl 1035.47015)] from one to many variables.It is shown that there are at most \(M_1 M_2 \cdots M_n\) different minimal reducing subspaces for \(M_{z_1}^{M_1}, M_{z_2}^{M_2}, \dots, M_{z_n}^{M_n}\), the precise number depending on the weight. Each of the minimal subspaces is generated by one polynomial.Each common reducing subspace \(X\) of \(M_{z_1}^{M_1}, M_{z_2}^{M_2}, \dots, M_{z_n}^{M_n}\) can be decomposed into minimal reducing subspaces by repeated solving of an extremal problem in \(X\). Reviewer: Michał Goliński (Poznań) Cited in 1 Document MSC: 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47A15 Invariant subspaces of linear operators 47A13 Several-variable operator theory (spectral, Fredholm, etc.) Keywords:reducing subspaces; weighted shift Citations:Zbl 1035.47015 × Cite Format Result Cite Review PDF Full Text: Euclid References: [1] A. Brown and P. R. Halmos, Algebraic Properties of Toeplitz Operators , J. Reine Angew. Math. 213 (1963), 89-102. · Zbl 0116.32501 [2] R. G. Douglas, Banach Algebra Techniques in Operator Theory. Second Edition , Graduate Texts in Mathematics 179 , Springer-Verlag, New York, 1998. · Zbl 0920.47001 [3] P. R. Halmos, Shifts on Hilbert spaces , J. Reine Angew. Math. 208 (1961), 102-112. [4] Yanyue Shi and Yufeng Lu, Reducing Subspaces for Toeplitz Operators on the polydisk , Bull. Korean Math. Soc. 50 (2013), 687-696. · Zbl 1280.47039 · doi:10.4134/BKMS.2013.50.2.687 [5] M. Stessin and K. Zhu, Reducing subspace of weighted shift operators , Proc. Amer. Math. Soc. 130 (2002), 2631-2639. · Zbl 1035.47015 · doi:10.1090/S0002-9939-02-06382-7 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.