On a Class of Operators of Finite Type

This paper studies some 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 self-commutator and invariant with respect to A*. Besides, in this class the spectrum of operator A is covered by the projection of a union of quadrature domains in some Riemann surfaces. In this paper the analytic model, the mosaic and some kernel related to the eigenfunctions are introduced which are the analogue of those objects in the theory of subnormal operators.