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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
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