Linear analysis of quadrature domains, III

The linear data of a quadrature domain, introduced in the first paper in th is series, is further investigated. The matricial structure of all hyponorm al operators with finite rank self-commutators, which is inductively constr ucted from such linear data, is discussed. In particular, a conservation la w (of a determinantal function) in the iterative process of defining these matrices is exploited. The spectrum of the determining left upper corner of some of these operators is analyzed in connection with the existence of ge neralized quadrature formulas on planar domains. (C) 1999 Academic Press.