The Grunsky operator and the Schatten ideals
type: Item
instance of: scholarly article
title: The Grunsky operator and the Schatten ideals
zbMATH Open document ID: 0968.47008
author: Gavin L. Jones
published in: Michigan Mathematical Journal
publication date: 2001-01-01T00:00:00Z
review text: Theorem 1. Let \(g\) be a conformal map of the unit disc to the interior of a Jordan curve. The Grunsky operator lies in the \(p\)th Schatten ideal \(\gamma_p\) \((p\geq 1)\) of operators on the Dirichlet space if and only if log \(g'\in B_p\), where \(B_p\) stands for the Besov space. Theorem 2. Let \(J\) be a quasicircle with welding \(h\) and let \(V_h\) be the composition operator \(f\rightarrow f\circ h\). Then the commutator \([V_h, H]\) of \(V_h\) with the Hilbert transform \(H\) lies in \(\gamma_p\) if and only if log \(g'\in B_p\), where \(g\) is the conformal map to the interior.
reviewed by: Bohumír Opic
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 1548295
zbMATH Keywords: Grunsky operator
zbMATH Keywords: Schatten ideals
zbMATH Keywords: Besov space
zbMATH Keywords: Jordan curve
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1030132361
OpenAlex ID: W2050349101
DOI: 10.1307/MMJ/1030132361
description: scientific article; zbMATH DE number 1548295