Linear analysis of quadrature domains (Q1915865)

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.