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.