Invariant subspaces for translation, dilation and multiplication semigroups
type: Item
instance of: scholarly article
title: Invariant subspaces for translation, dilation and multiplication semigroups
zbMATH Open document ID: 1190.47005
author: Eva A. Gallardo Gutiérrez
author: Jonathan R. Partington
published in: Journal d'Analyse Mathématique
publication date: 2009-12-14T00:00:00Z
review text: Considering the right-shift semigroup \(\{S_\tau\}_{\tau\geq 0}\) and the composition operator \(C_{1/\lambda}\) defined by \(C_{1/\lambda}f(x)=f(x/\lambda)\) for \(f\in L^2(\mathbb R)\), a characterization of the common invariant subspaces is obtained. For any given hyperbolic automorphism \(\varphi\) of the unit disc \(\mathbb D\), the lattice of the invariant subspaces of the composition operator \(C_\varphi\) on \(H^2(\mathbb D)\) strictly contains \(\text{Lat}(S_\tau, C_{1/\lambda})\), with \(0< \lambda< 1\). In particular, it is shown that the elements of \(\text{Lat}(C_{1/\lambda})\) in \(L^2(0,\infty)\) are in one-one correspondence with those of the bilateral shift on a vector valued space. Also for \(M_a\), the multiplication by \(e^{iax}\), the lattice of jointly invariant subspaces \(\text{Lat}(S_\tau, M_a)\) is characterized. An application to the study of the lattice of invariant subspaces for composition operators induced by parabolic automorphism of the unit disc \(\mathbb D\) is provided.
reviewed by: Ilie Valuşescu
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 5646612
zbMATH Keywords: right-shift semigroup
zbMATH Keywords: common invariant subspaces for translation
zbMATH Keywords: invariant subspace problem
MaRDI profile type: Publication
full work available at URL: s11854-009-0003-6
OpenAlex ID: W2086998512
Wikidata QID: Q56319250
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DOI: 10.1007/S11854-009-0003-6
description: scientific article; zbMATH DE number 5646612