Linear analysis of quadrature domains
type: Item
instance of: scholarly article
title: Linear analysis of quadrature domains
zbMATH Open document ID: 0892.47025
DOI: 10.1007/BF02559714
author: Mihai Putinar
published in: Arkiv för Matematik
publication date: 1998-08-03T00:00:00Z
review text: A bounded planar domain \(U\) is called a quadrature domain, if there exists a distribution \(u\) with finite support in \(U\), such that \(\int_U fdA= u(f)\), for every integrable analytic function \(f\) in \(U\) (and \(dA\) stands for the area measure). See for details: \textit{H. S. Shapiro}, ``The Schwarz function and its generalization to higher dimensions'' (1992; Zbl 0784.30036). Helton-Howe's formula in the theory of semi-normal operators was used as a link between quadrature domains and a class of operators with rank-one self-commutator, see the author, J. Funct. Anal. 136, 331-364 (1996). These operators are determined by a compression to a semi-invariant subspace of finite dimension. Therefore the class of quadrature domains is parametrized by a finite matrix. The present paper contains the precise transition formulae between a quadrature domain and its associated matrix. In particular, the quadrature identity, a defining function for the domain, the Schwarz function have simple expression in terms of this linear parameter.
reviewed by: Mihai Putinar
Mathematics Subject Classification ID: 47B20
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 47A57
zbMATH DE Number: 894906
zbMATH Keywords: bounded planar domain
zbMATH Keywords: quadrature domain
zbMATH Keywords: distribution
zbMATH Keywords: integrable analytic function
zbMATH Keywords: semi-normal operators
zbMATH Keywords: operators with rank-one self-commutator
zbMATH Keywords: compression to a semi-invariant subspace of finite dimension
zbMATH Keywords: associated matrix
zbMATH Keywords: quadrature identity
zbMATH Keywords: Schwarz function
MaRDI profile type: Publication
cites work: Domains on which analytic functions satisfy quadrature identities
cites work: Q4776444
cites work: Commutators, symbols, and determining functions
cites work: Seminormal operators
cites work: A kernel for operators with one-dimensional self-commutators
cites work: Q3721245
cites work: Q3715697
cites work: Q4044857
cites work: Quadrature identities and the Schottky double
cites work: Singular and special points on quadrature domains from an algebraic geometric point of view
cites work: Traces of commutators of integral operators
cites work: Commutators and systems of singular integral equations. I
cites work: The analytic model of a hyponormal operator with rank one self-commutator
cites work: Extremal solutions of the two-dimensional \(L\)-problem of moments
cites work: Quadrature domains
cites work: Q4004082
cites work: On the kernels associated with a class of hyponormal operators
full work available at URL: https://doi.org/10.1007/bf02559714
OpenAlex ID: W2162840356
description: scientific article; zbMATH DE number 894906