Equivalence of Hardy submodules generated by polynomials
type: Item
instance of: scholarly article
title: Equivalence of Hardy submodules generated by polynomials
zbMATH Open document ID: 0977.46028
author: Kunyu Guo
published in: Journal of Functional Analysis
publication date: 2002-01-07T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 46H25
zbMATH DE Number: 1563142
zbMATH Keywords: Hardy space
zbMATH Keywords: Hardy submodule
zbMATH Keywords: polydisk
zbMATH Keywords: Unitary equivalence
zbMATH Keywords: similarity
zbMATH Keywords: quasisimilarity
reviewed by: Timothy G. Feeman
MaRDI profile type: Publication
full work available at URL: jfan.2000.3659
OpenAlex ID: W2076705680
cites work: Invariant subspaces in the polydisk
cites work: Q5611963
cites work: Q5563439
cites work: Q4009889
cites work: Q3992983
cites work: Algebraic Reduction and Rigidity for Hilbert Modules
cites work: On the rigidity of Hardy submodules
cites work: Q3922907
cites work: Rings with Topologies Induced by Spaces of Functions
cites work: Topological Hilbert Nullstellensatz for Bergman spaces
cites work: Characteristic spaces and rigidity for analytic Hilbert modules
cites work: Q4529349
cites work: Q4248867
cites work: Holomorphic functions of several variables. An introduction to the fundamental theory. With the assist. of Gottfried Barthel transl. by Michael Bridgland
cites work: Q3924489
cites work: Q5567211
cites work: Q3958740
cites work: Q4695258
cites work: Q4529380
cites work: Q5732721
DOI: 10.1006/JFAN.2000.3659
description: scientific article; zbMATH DE number 1563142