Rank-one commutators on invariant subspaces of the Hardy space on the bidisk
type: Item
instance of: scholarly article
title: Rank-one commutators on invariant subspaces of the Hardy space on the bidisk
zbMATH Open document ID: 1099.47003
author: Kou-Hei Izuchi
author: Kei-Ji Izuchi
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-03-29T00:00:00Z
review text: Let \(H^2(\Gamma^2)\) be the Hardy space on the two-dimensional torus on \(\Gamma^2\). Let \(T_z\) and \(T_w\) be the multiplication operators on \(H^2(\Gamma^2)\) by the coordinate functions \(z\) and \(w\), respectively. Let \(M\) be an invariant subspace in \(H^2(\Gamma^2)\), that is, \(T_zM\subset M\) and \(T_w M\subset M\). For an invariant subspace \(M\) of \(H^2(\Gamma^2)\), put \(V_w= P_M T_z\) and \(V_w= P_M T_w\) on \(M\), where \(P_M\) is the orthogonal projection onto \(M\). Write \([V_w, V^*_z]= V_w V^*_z- V^*_z V_w\). The authors describe an invariant subspace \(M\) in \(H^2(\Gamma^2)\) such that \([V_m, V^*_z]\) is a nonzero selfadjoint operator and they show that \(\text{rank}[V_w, V^*_z]= 1\). Moreover, they study some examples of invariant subspaces in \(H^2(\Gamma^2)\) such that \(\text{rank}[V_w, V^*_z]<\infty\).
reviewed by: Takahiko Nakazi
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 46E20
zbMATH DE Number: 5016162
zbMATH Keywords: invariant subspace
zbMATH Keywords: rank-one operator
zbMATH Keywords: Hardy space on the bidisk
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.04.021
OpenAlex ID: W2046252371
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DOI: 10.1016/J.JMAA.2005.04.021
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description: scientific article; zbMATH DE number 5016162