The Dirac operator of a commuting d-tuple

Abstract

Given a commuting d-tuple $\bar T=(T_1,...,T_d)$ of otherwise arbitrary nonnormal operators on a Hilbert space, there is an associated Dirac operator $D_{\bar T}$. Significant attributes of the d-tuple are best expressed in terms of $D_{\bar T}$, including the Taylor spectrum and the notion of Fredholmness. In fact, {\it all} properties of $\bar T$ derive from its Dirac operator. We introduce a general notion of Dirac operator (in dimension $d=1,2,...$) that is appropriate for multivariable operator theory. We show that every abstract Dirac operator is associated with a commuting $d$-tuple, and that two Dirac operators are isomorphic iff their associated operator $d$-tuples are unitarily equivalent. By relating the curvature invariant introduced in a previous paper to the index of a Dirac operator, we establish a stability result for the curvature invariant for pure d-contractions of finite rank. It is shown that for the subcategory of all such $\bar T$ which are a) Fredholm and and b) graded, the curvature invariant $K(\bar T)$ is stable under compact perturbations. We do not know if this stability persists when $\bar T$ is Fredholm but ungraded, though there is concrete evidence that it does.