(c) 1993 Birkh~user Verlag, Basel TRACE FORMULAS FOR A CLASS OF SUBNORMAL TUPLES OF OPERATORS

Trace formulas are established for the product of commutators related to subnormal tuple of operators ($1,..., Sn) with minimal normal extension (N1, 9 9 9 N~) satisfying conditions that sp(Sy)\sp(Ny) is simply-connected with smooth boundary Jordan curve sp(Ni) and [S~, Sy] 1/2 9 1, j = 1, 2,..., n. Some complete unitary invariants related to the trace formulas are found. 1. Let ~/be a complex separable Hilbert space, /~(~) be the algebra of linear bounded operator on) / and let ~1()/) be the trace ideal of/]() /). For T E L(~/), let CU(T) be a set of objects determined by T. The set CU(.) is said to be a complete unitary invariant for the operator T in a class if for every pair of operators S and T in this class CU(T) = CU(S) is a necessary and sufficient condition for the existence of a unitary operator U satisfying S = UTU-1. A basic problem in operator theory is to find out some simple complete unitary invariant for operator on a Hilbert space. There are many different approaches such as [AD], [Ca] etc. In this paper, we study complete unitary invariants related to some trace formulas for some subnormal tuples of operators