Trace Formulas for Some Operators Related To Quadrature Domains in Riemann Surfaces

This note studies the trace formula of a class of pure operators A with finite rank self-commutators satisfying the condition that there is a finite dimensional subspace containing the image of the selfcommutator and invariant with respect to A *. Besides, in this class, the spectrum of the operator A is covered by the projection of a union of quadrature domains in some Riemann surfaces.