A Note on Hyponormal Operators Associated with Quadrature Domains

Abstract

In a series of papers [15], [16], [10], Putinar and Gustafsson connect the properties of a quadrature domain D with the properties of the hyponormal operator T with rank one self-commutator, whose principal function coincides with the characteristic function of this domain. Xia’s analytic model [19] of this hyponormal operator represents it as the operator of multiplication by the independent variable on a space of analytic functions on D. We give a new formula for the norm in this model space. The relationship between quadrature domains and subnormal operators with finite rank self-commutator has been found and exploited in [20], [12], [23]. Our result shows that a hyponormal T as above can be obtained from a subnormal operator with finite rank self-commutator by a finite rank perturbation of the space and the norm in it.