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On quotient modules of \(H^2( \mathbb{D}^n)\): essential normality and boundary representations. (English) Zbl 1445.47006

Let \(H^2(\mathbb{D}^n)\) be the Hardy space of holomorphic functions on the unit polydisc \(\mathbb{D}^n\), \(n\geqslant 1\). The space \(H^2(\mathbb{D}^n)\) is also understood as a module over \(\mathbb{C}[z_1,\ldots,z_n]\). Let \((M_{z_1},\dots, M_{z_n})\) be the \(n\)-tuple of multiplication operators by the coordinate functions on \(H^2(\mathbb{D}^n)\). A closed subspace \(\mathcal{S}\) of \(H^2(\mathbb{D}^n)\) is called a submodule if \(M_{z_i}\mathcal{S}\subset \mathcal{S}\) for all \(i=1,\dots,n\). A closed subspace \(\mathcal{Q}\) of \(H^2(\mathbb{D}^n)\) is a quotient module if \(\mathcal{Q}^\bot(\cong H^2(\mathbb{D}^n)/\mathcal{Q})\) is a submodule. A quotient module \(\mathcal{Q}\) is said to be of Beurling type if \(\mathcal{Q}=H^2(\mathbb{D}^n)\ominus \theta H^2(\mathbb{D}^n)\cong H^2(\mathbb{D}^n)/\theta H^2(\mathbb{D}^n)\) for some inner function \(\theta\in H^\infty(\mathbb{D}^n)\). A quotient module \(\mathcal{Q}\) of \(H^2(\mathbb{D}^n)\) is essentially normal if the commutator \([C_{z_i},C^\ast_{z_j}]\) is compact for all \(1\leqslant i,j\leqslant n\), where \(C_{z_i}\) is the compression of the operator \(M_{z_i}\) to the subspace \(\mathcal{Q}\).
The obtained results are described by the authors as follows: “In this paper, we first investigate the essential normality of certain classes of quotient modules including Beurling-type quotient modules of \(H^2(\mathbb{D}^n)\), \(n\geqslant 3\). We prove that the Beurling type quotient modules of \(H^2(\mathbb{D}^n)\) \((n\geqslant 3)\) and the Rudin quotient modules of \(H^2(\mathbb{D}^2)\) are not essentially normal. We obtain a complete characterization for essential normality of doubly commuting quotient modules of an analytic Hilbert module (defined in paragraph 2) over \(\mathbb{C}[z]\) including \(H^2(\mathbb{D}^n)\) and the weighted Bergman modules \(L^2_{a,\alpha}(\mathbb{D}^n)\) \((\alpha\in\mathbb{Z}^n\), \(\alpha_i>-1\), \(i=1,\dots,n)\) as special cases \((n\geqslant 2)\).
“We study boundary representations for doubly commuting quotient modules of an analytic Hilbert module over \(\mathbb{C}[z]\), and obtain some direct results for the case \(H^2(\mathbb{D}^n)\) and \(L^2_a(\mathbb{D}^n)\) \((n\geqslant 2)\) (see Corollaries 4.3 and 4.4). We also consider the class of homogeneous quotient modules of \(H^2(\mathbb{D}^2)\).”

MSC:

47A13 Several-variable operator theory (spectral, Fredholm, etc.)
47A20 Dilations, extensions, compressions of linear operators
47L25 Operator spaces (= matricially normed spaces)
47L40 Limit algebras, subalgebras of \(C^*\)-algebras
46L05 General theory of \(C^*\)-algebras
46L06 Tensor products of \(C^*\)-algebras
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