Subnormal operators of finite type II. Structure theorems

Abstract

This paper concerns pure subnormal operators with finite rank self-commutator, which we call subnormal operators of finite type. We analyze Xia's theory of these operators [21–23] and give its alternative exposition. Our exposition is based on the explicit use of a certain algebraic curve in ${\mathbb{C}}^2$, which we call the discriminant curve of a subnormal operator, and the approach of dual analytic similarity models of [26]. We give a complete structure result for subnormal operators of finite type which corrects and strenghtens the formulation that Xia gave in [23]. Xia claimed that each subnormal operator of finite type is unitarily equivalent to the operator of multiplication by $z$ on a weighted vector $H^2$-space over a “quadrature Riemann surface” (with a finite rank perturbation of the norm. We explain how this formulation can be corrected and show that, conversely, every “quadrature Riemann surface” gives rise to a family of subnormal operators. We prove that this family is parametrized by the so-called characters. As a departing point of our study, we formulate a kind of scattering scheme for normal operators, which includes Xia's model as a particular case.