Complete unitary invariant for some subnormal operator

Abstract

IfS is a subnormal operator with minimal normal extensionN satisfying the conditions that (i)$\left[ {S^* ,S} \right]^{\frac{1}{2}} \in \mathcal{L}^1$, (ii) sp (S) is the unit disk and (iii) sp (N)={N: |z|=1 orz=a ₁,...,a _k then

$$tr\left( {\left[ {S^* ,(\lambda I - S)^{ - 1} } \right]\left[ {S^* ,(\mu I - S)^{ - 1} } \right]} \right) = \frac{n}{{\lambda ^2 \mu ^2 }} + \sum\limits_{i,j = 1}^k {\frac{{\gamma ij}}{{\lambda \mu (\lambda - a_i )(\mu - a_j )}}} $$

. wheren=index ($S^* - \bar zI$) forz∈sp (S)/sp (N) and (γij) is a real symmetric matrix. The set {n, γij,i,j = 1,...,k} is a complete unitary invariant for an operator in the class of all irreducible subnormal operators satisfying conditions (i), (ii), (iii) and that there is at least one positive simple eigenvalue of [S ^*,S].