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Summary: THE DIRAC OPERATOR OF A COMMUTING d-TUPLE
William Arveson
Department of Mathematics
University of California
Berkeley CA 94720, USA
Abstract. Given a commuting d-tuple ¯T = (T1, . . . , Td) of otherwise arbitrary op-
erators on a Hilbert space, there is an associated Dirac operator D ¯T . Significant
attributes of the d-tuple are best expressed in terms of D ¯T , including the Taylor
spectrum and the notion of Fredholmness.
In fact, all properties of ¯T derive from its Dirac operator. We introduce a general
notion of Dirac operator (in dimension d = 1, 2, . . . ) that is appropriate for multivari-
able 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 ¯T which
are a) Fredholm and and b) graded, the curvature invariant K( ¯T) is stable under
compact perturbations. We do not know if this stability persists when ¯T is Fredholm
but ungraded, though there is concrete evidence that it does.
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