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The operator \(M_{z_1^n z_2^n}\) on subspaces of Bergman spaces over the biannulus. (English) Zbl 1443.47026

The authors regard the set \(A_{r_j}\) as a subset of the unit disk and the bi-annulus \(A_r^2\) as a subset of \(\mathbb{C}^2\) corresponding to \(r=(r_1,r_2)\) for \(0< r_1 \neq r_2<1\). For any \(A_r^2\) and any \(\alpha=(\alpha_1,\alpha_2)\) such that \(\alpha_1,\alpha_2>-1\), they define a weighted Bergman space \(L^2(A_r^2, dv_{\alpha})\) and show that this space is a reproducing kernel Hilbert space. In the following, they define the closed subspace \(L_{\alpha,\alpha}^{2++}(A_r^2)\) of \(L^2(A_r^2, dv_{\alpha})\) and consider the multiplication operator \(M_{z_1^nz_2^n}\) on that space.
Finally, they study some properties of the operator \(M_{z_1^nz_2^n}\) and verify the similarity of \(M_{z_1^nz_2^n}\) and \(\bigoplus_1^{n^2}M_{z_1z_2}\) on \(L_{\alpha,\alpha}^{2++}(A_r^2)\).

MSC:

47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
32A36 Bergman spaces of functions in several complex variables
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