Trace formulas and \(p\)-essentially normal properties of quotient modules on the bidisk
type: Item
instance of: scholarly article
zbMATH Open document ID: 1261.47014
author: Kunyu Guo
author: Kai Wang
author: Genkai Zhang
publication date: 2012-06-13T00:00:00Z
Mathematics Subject Classification ID: 47A13
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 46C99
zbMATH DE Number: 6045979
zbMATH Keywords: Hilbert module
zbMATH Keywords: quotient module
zbMATH Keywords: essentially normal quotient
zbMATH Keywords: trace class
zbMATH Keywords: Hilbert-Schmidt class
zbMATH Keywords: Dirichlet norm
MaRDI profile type: Publication
title: Trace formulas and \(p\)-essentially normal properties of quotient modules on the bidisk
published in: Journal of Operator Theory
review text: Let \(H=H_\nu\) be the functional Hilbert space on the unit disc \(\mathbb D\) with the reproducing kernel \(K_w(z)=\frac{1}{(1-z\bar{w})^\nu}\) for \(\nu\geq 1\). It is the Hardy space \(H^2(\mathbb T)\;(\nu=1)\) or the weighted Bergman space \(L^2_a(\mathbb D,d\mu_{\nu-2})\) \((\nu>1)\), where \(d\mu_\alpha=c_\alpha(1-| z|^2)^\alpha dm(z)\) is the normalized measure on \(\mathbb D\). The space \(H\otimes H\) is then the Hilbert space \(H^2(\mathbb T^2)\) or \(L^2_a(\mathbb D^2, d\mu_{\nu-2}\times d\mu_{\nu-2})\) on the bidisc \(\mathbb D^2\). Let \(M_f\) be the multiplication operator on \(H\otimes H\) for \(f\in H^\infty(\mathbb D)\). For an invariant subspace \(\mathcal M\) of the multiplication operators \(M_z, M_w\) on \(H\otimes H\) denote by \(S_f=P_{\mathcal M^\bot}M_f P_{\mathcal M^\bot}\) and \(H_f=P_{\mathcal M}M_fP_{\mathcal M^\bot}\), respectively, the compression of \(M_f\) to the quotient module \(\mathcal M^\bot\) and the Hankel type operator. The authors study properties of commutators \([S^\ast_f,S_f]\) and, among others, prove the following:NEWLINENEWLINE \noindent Theorem 1. The commutators \([S^\ast_z,S_z]\), \([S^\ast_w,S_w]\), \([S^\ast_z,S_w]\) are of trace class. Thus the quotient module \(\mathcal M^\bot\) is in \(\mathcal L^1\) (trace class operators).NEWLINENEWLINE They obtain the following trace formula:NEWLINENEWLINE \noindent Theorem 2. Let \(F(z,w)\) be a polynomial in \((z,w)\) and \(f(z)=F(z,z)\) be its restriction to the diagonal. Then NEWLINE\[NEWLINE{\text{Tr}}\,[S^\ast_F,S_F]=(N+1)\int_{\mathbb D}| f'(z)|\,dm(z).NEWLINE\]NEWLINE Similar trace formulas for Hankel type operators are also obtained.
reviewed by: Andrzej Sołtysiak
description: scientific article; zbMATH DE number 6045979