Positive Toeplitz operators of finite rank on the parabolic Bergman spaces
type: Item
instance of: scholarly article
title: Positive Toeplitz operators of finite rank on the parabolic Bergman spaces
zbMATH Open document ID: 1310.47044
author: Masaharu Nishio
author: Noriaki Suzuki
author: Masahiro Yamada
published in: Kodai Mathematical Journal
publication date: 2013-05-24T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.kmj/1364562716
review text: Let \({\mathbb R}_+^{n+1}={\mathbb R}^n\times(0,\infty)\), \(0<\alpha\leq 1\), \(\lambda>-1\), and \(L^{(\alpha)}=\partial/\partial t+(-\partial/\partial x_1-\dots-\partial/\partial x_n)^\alpha\) for \((x_1,\dots,x_n,t)\in{\mathbb R}_+^{n+1}\). Let \(b_\alpha^2(\lambda)\) denote the parabolic Bergman space consisting of all \(L^{(\alpha)}\)-harmonic functions on \({\mathbb R}_+^{n+1}\) that are square integrable with respect to the weighted measure \(t^\lambda\,dx\,dt\) on \({\mathbb R}_+^{n+1}\). For a nonnegative Radon measure \(\mu\) on \({\mathbb R}_+^{n+1}\), let \(\widetilde{T}_\mu^\lambda\) be the corresponding Toeplitz operator. The main result of the paper says that, if there exists a dense subspace \({\mathcal D}_0\) in \(b_\alpha^2(\lambda)\) such that \({\mathcal D}_0\subset\text{Dom}(\widetilde{T}_\mu^\lambda)\) and \(\dim(\widetilde{T}_\mu^\lambda({\mathcal D}_0))<\infty\), then \(\mu\) is a finite linear combination of point masses and \(\text{rank}(\widetilde{T}_\mu^\lambda)=\#\text{supp}(\mu)=\dim(\widetilde{T}_\mu^\lambda({\mathcal D}_0))\), where \(\# A\) denotes the cardinal number of a set \(A\).
reviewed by: Oleksiy Karlovych
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 31B10
zbMATH DE Number: 6165609
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: finite rank operator
zbMATH Keywords: parabolic Bergman spaces
MaRDI profile type: Publication
OpenAlex ID: W2020693775
DOI: 10.2996/KMJ/1364562716
description: scientific article; zbMATH DE number 6165609