Complete unitary invariant for some subnormal operator

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