A Commutator Formula for Subnormal Tuples of Operators

Let S = (S1,..., Sk) be a pure subnormal k- tuple of operators with minimal normal extension N and defect space M. Let. j = (S* j | M)*. We prove where e(.) = PME(.)(M), E(.) is the spectral measure of N, PM is the projection to M, f is any analytic function on x(j)(k) = 1 sigma(S-j*) and h is any analytic function on x(j)(k) = 1(sigma)s(S-j). If dim M < infinity, then this commutator equals to 1/2 pi i integral(A) mu j(uj) df(<(u)over bar>)dh(u), where A = {((u) over bar,...,(u) over bark) : u = (u1,..., uk) is in the joint point spectrum of S*}, and mu j(.) is the mosaic of S-j. Besides, a similar commutator formula for a pure hyponormal operator associated with a quadrature domain is established.