  Consider a Hilbert space obtained as the completion of the polynomials C[z}
in m-variables for which the mnonomials are orthogonal. If the commuting
weighted shifts defined by the coordinate functions are essentially normal,
then the same is true for their restrictions to invariant subspaces spanned by
monomials. This generalizes the result of Arveson [4] in which the Hilbert
space is the m-shift Hardy space H_m^2. He establishes his result for the case
of finite multiplicity and shows the self-commutators lie in the Schatten
p-class for p > m. We establish our result at the same level of generality. We
also discuss the K-homology invariant defined in these cases.
