Note on the Curvature and Index of Almost Unitary Contraction Operator *

In the recent preprint [1] S. Parrott proves the equality between the Arveson's curvature and the Fredholm index of a " pure " contraction with finite defect numbers. In the present note one derives a similar formula in the " non-pure " case. The notions of d-contraction T = (T 1 , T 2 ,. .. , T d) and its curvature was introduced by W. Arveson in a series of papers (see [2], [3], and [4]). In the case of a single contraction (d = 1) the curvature is thoroughly investigated in the paper of Parrott [1]. Namely, let T be a contraction operator on a Hilbert space H, and suppose that ∆ T = √ 1 − T T * has finite rank 1. Parrott shows that the curvature K(T) of T can be defined on three equivalent ways: K(T) = |z|=1 dz lim r↑1 (1 − r 2) tr (∆ T (1 − rzT *) −1 (1 − r¯ zT) −1 ∆ T) = lim n→∞ tr (1 − T n T * n) n = lim n→∞ tr (T * n T n (1 − T T *)). In the papers cited above Arveson introduces the notion of " pure " d-contraction. In the case of a single contraction T this reduces to the condition that T belongs to the class C ·,0 , i.e for any h ∈ H we have T 1 Most of the results of [1] still hold in the case when ∆ T belongs to the trace class.