Tensor product of quotient Hilbert modules. (English) Zbl 1302.47034
Summary: In this paper, we present a unified approach to problems of tensor product of quotient modules of Hilbert modules over \(\mathbb{C} [z]\) and corresponding submodules of reproducing kernel Hilbert modules over \(\mathbb{C} [z_1, \dots, z_n]\) and the doubly commutativity property of module multiplication operators by the coordinate functions. More precisely, for a reproducing kernel Hilbert module \(\mathcal{H}\) over \(\mathbb{C} [z_1, \dots, z_n]\) of analytic functions on the polydisc in \(\mathbb{C}^n\) which satisfies certain conditions, we characterize the quotient modules \(\mathcal{Q}\) of \(\mathcal{H}\) such that \(\mathcal{Q}\) is of the form \(\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n\), for some one-variable quotient modules \(\{\mathcal{Q}_1, \dots, \mathcal{Q}_n \}\). For \(\mathcal{H}\) the Hardy module over the polydisc, \(H^2(\mathbb{D}^n)\), this reduces to some recent results by K. Izuchi et al. [J. Oper. Theory 51, No. 2, 361–376 (2004; Zbl 1055.47009)], and the third author [ibid. 72, 371–385 (2014), arxiv:1303.1041]. This is used to obtain a classification of co-doubly commuting submodules for a class of reproducing kernel Hilbert modules over the unit polydisc. These results are applied to compute the cross commutators of co-doubly commuting submodules. Moreover, this provides further insight into the wandering subspaces and ranks of submodules of the Hardy module. Our results include the case of weighted Bergman modules over the unit polydisc in \(\mathbb{C}^n\).
MSC:
| 47B32 | Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) |
| 47A15 | Invariant subspaces of linear operators |
| 46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |
| 46H25 | Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) |
Keywords:
Hilbert modules; Hardy and weighted Bergman spaces over the polydisc; doubly commuting quotient modules; essential normality; wandering subspace; rankCitations:
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