Beurling's phenomenon in two variables
type: Item
instance of: scholarly article
title: Beurling's phenomenon in two variables
zbMATH Open document ID: 1061.46023
author: Rongwei Yang
published in: Integral Equations and Operator Theory
publication date: 2004-09-22T00:00:00Z
review text: The author extends the Beurling's phenomenon from the Hardy space over the unit disk \(H^2(\mathbb{D}^2)\) to the two variable Hardy space \(H^2(\mathbb{D})\). Although in \(H^2(\mathbb{D}^2)\) things are far more complicated and there is no similar characterization of invariant subspaces in terms of inner functions as in \(H^2 (\mathbb{D})\), the author shows in this paper an analogous phenomenon in terms of reproducing kernels. It is well-known that every shift-invariant subspace \(M\) is of the form \(\psi H^2 (\mathbb{D})\) for some inner function \(\psi\) by Beurling's theorem, and the reproducing kernel of \(M\) is \(K_\lambda^M(z)=\frac {\overline {\psi (\lambda)}\psi(z)} {1-\overline\lambda z}\). Since \((1-|\lambda|^2) K_\lambda^M (\lambda)= |\psi(\lambda)|^2\), the fact that \(\psi\) is inner is equivalent to the property that \((1-|\lambda|^2) K_\lambda^M (\lambda)\) has boundary value 1 almost every where on \(\partial\mathbb{D}\), and so \((1-|\lambda|^2) K_\lambda^M(\lambda)\) is subharmonic. For the two variable space \(H^2(\mathbb{D}^2)\), a natural analogue of \((1-|\lambda|^2)K_\lambda(\lambda)\) is \(G^M(\lambda_1, \lambda_2)=(1-|\lambda_1|^2) (1-|\lambda_2|^2)K^M_{(\lambda_1,\lambda_2)} (\lambda_1,\lambda_2)\), where \(K^M_\lambda(z)\) is the reproducing kernel of \(M\). In this paper, the author shows that \(G^M(\lambda_1, \lambda_2)\) is subharmonic in each variable, and under a mild condition it is shown that \(G^M(\lambda_1, \lambda_2)=1\) almost everywhere on \(\partial\mathbb{D}^2\).
Mathematics Subject Classification ID: 46E20
Mathematics Subject Classification ID: 32A50
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 46E22
zbMATH DE Number: 2101687
zbMATH Keywords: Beurling's theorem
zbMATH Keywords: reproducing kernel
zbMATH Keywords: \(H^2(\mathbb{D}^2)\)
zbMATH Keywords: Berzin transform
zbMATH Keywords: trace class operator
zbMATH Keywords: subharmonic function
reviewed by: Bahmann Yousefi
MaRDI profile type: Publication
full work available at URL: s00020-002-1195-7
OpenAlex ID: W1993235001
DOI: 10.1007/S00020-002-1195-7
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description: scientific article; zbMATH DE number 2101687