p-SUMMABLE COMMUTATORS IN DIMENSION d

We show that many invariant subspaces M for d-shifts (S1, . . . , Sd) of finite rank have the property that the orthogonal projection PM onto M satisfies PMSk − SkPM ∈ L, 1 ≤ k ≤ d for every p > 2d, L denoting the Schatten-von Neumann class of all compact operators having p-summable singular value lists. In such cases, the d tuple of operators T̄ = (T1, . . . , Td) obtained by compressing (S1, . . . , Sd) to M ⊥ generates a ∗-algebra whose commutator ideal is contained in L for every p > d. It follows that the C∗-algebra generated by {T1, . . . , Td} and the identity is commutative modulo compact operators, the Dirac operator associated with T̄ is Fredholm, and the index formula for the curvature invariant is stable under compact perturbations and homotopy for this restricted class of finite rank d-contractions. Though this class is limited, we conjecture that the same conclusions persist under much more general circumstances.