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# The Dirac operator of a commuting d-tuple

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### 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.

### Most cited references3

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### Subalgebras of C*-algebras III: Multivariable operator theory

(1998)
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### Fredholm and invertible $n$-tuples of operators. The deformation problem

(1981)
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### The stability of the Euler characteristic for Hilbert complexes

(1980)
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### Author and article information

###### Journal
2000-05-30
2000-06-11
math/0005285