Subnormal operators and quadrature domains
type: Item
instance of: scholarly article
title: Subnormal operators and quadrature domains
zbMATH Open document ID: 0902.47024
author: John E. McCarthy
author: L. M. Yang
published in: Advances in Mathematics
publication date: 1998-12-09T00:00:00Z
review text: A bounded linear operator \(S\) on a Hilbert space \({\mathcal H}\) is called subnormal if it has a normal extension. The self-commutator of \(S\) is \([S^*,S]= S^*S- SS^*\). The operator \(S\) is called cyclic if there exists a vector \(x\in {\mathcal H}\) such that \(\{p(S)x:p\) is a polynomial\} is dense in \({\mathcal H}\), and is called rationally cyclic if there exists \(x\) so that \(\{r(S)x:r\) is a rational function with poles off \(\sigma(S)\}\) is dense in \({\mathcal H}\) where \(\sigma(S)\) denotes the spectrum of \(S\). An operator is called irreducible if it commutes with no non-trivial projection. It is known, by Olin, Thomson and Trent, the necessary and sufficient condition that an irreducible cyclic subnormal operator \(S\) has a finite rank self-commutator. The main result in this paper is a generalization of this result in the case where \(S\) is rationally cyclic.
Mathematics Subject Classification ID: 47B20
zbMATH DE Number: 1013823
zbMATH Keywords: quadrature domains
zbMATH Keywords: normal extension
zbMATH Keywords: irreducible cyclic subnormal operator
zbMATH Keywords: finite rank self-commutator
zbMATH Keywords: rationally cyclic
reviewed by: Takashi Yoshino
MaRDI profile type: Publication
full work available at URL: aima.1997.1619
OpenAlex ID: W2069523929
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DOI: 10.1006/AIMA.1997.1619
description: scientific article; zbMATH DE number 1013823