Equivalence of Hardy submodules generated by polynomials
For the Hardy space \(H^2\) of analytic square-integrable functions on the open unit disc, Beurling's theorem asserts that, if \(M\) is a subspace of \(H^2\) that is invariant under multiplication by \(z\), then there exists an inner function \(g\) in \(H^2\) such that \(M=gH^2\). In the multivariate context of functions on \(\mathbb{C}^n\), the appropriate generalization is to look at the structure of analytic submodules of either the Hardy space on the polydisk or the Hardy space of the unit ball. The polydisk is defined by \(D^n= \{z=(z_1,\ldots,z_n)\in\mathbb{C}^n : |z_j|<1 \forall j\}\) while the unit ball is defined by \(B_n=\{z=(z_1,\ldots,z_n)\in\mathbb{C}^n : {z_1}^2 + \cdots + {z_n}^2<1\}\). The boundary of \(B_n\) is the unit sphere while part of the boundary of the polydisk consists of the torus \(T^n=\{z\in\mathbb{C}^n : |z_j|=1 \forall j\}\). Two submodules, \(M_1\) and \(M_2\), of \(H^2(D^n)\) are said to be unitarily equivalent (respectively, similar) if there is a unitary (respectively, invertible) operator \(X:M_1\rightarrow M_2\) such that \(X(ph)=pX(h)\) for all polynomials \(p\) and all \(h\) in \(M_1\). Unitary equivalence, similarity, and a weaker notion called quasi-similarity have been studied by a variety of authors, including Douglas, Paulsen, K. Yan, Agrawal, and Clark, as well as the author of the paper reviewed here. For an ideal \(I\) of \(H^2(D^n)\) that is generated by finitely many polynomials (in \(n\) variables), let \([I]\) be the submodule of \(H^2(D^n)\) generated by \(I\). In fact, \([I]\) is the closure of \(I\). Moreover, taking \(p\) to be the greatest common divisor of the generators of \(I\), we can write \(I=pL\) where the ideal \(L\) has g.c.d. 1. This is referred to as the Beurling form of \(I\). The heart of the paper is the valuable and difficult Theorem 3.1, whose proof requires numerous lemmas and even a result about finitely generated analytic submodules that uses sheaf theory in its proof. Connections to earlier work of Guo, of Douglas and Paulsen, and of others are frequent and clearly addressed. The Theorem itself shows that, for two ideals \(I_1=p_1L_1\) and \(I_2=p_2L_2\), the submodules \([I_1]\) and \([I_2]\) of \(H^2(D^n)\) are unitarily equivalent if, and only if, \([p_1L_1]=[p_1L_2]\) and there exist polynomials \(q_1\) and \(q_2\) with no zeroes in \(D^n\) such that \(|p_1q_1|=|p_2q_2|\) on \(T^n\). Similarity of submodules is taken up in section 4, where the main result (Theorem 4.3) is that two homogeneous ideals \(I_1=p_1L_1\) and \(I_2=p_2L_2\) are similar if, and only if, they are quasi-similar. Moreover, each of these conditions is shown to be equivalent to the condition that \(L_1=L_2\) together with the existence of a constant \(c\) satisfying \(c<|p_1|/|p_2|<c^{-1}\) on the torus \(T^n\). This strengthens a result of K. Yan. The final section of the paper briefly considers submodules of \(H^2(B_n)\) showing that, in the case of homogeneous ideals \(I_1\) and \(I_2\), the unitary equivalence, similarity, and quasi-similarity of the submodules \([I_1]\) and \([I_2]\) are all equivalent to the condition that \(I_1=I_2\). The author concludes with the conjecture that, for ideals \(I_1\) and \(I_2\), the generated submodules \([I_1]\) and \([I_2]\) of \(H^2(B_n)\) are quasi-similar if, and only if, they are equal.
scientific article; zbMATH DE number 1563142
scientific article