THE DIRAC OPERATOR OF A COMMUTING d-TUPLE

Given a commuting d-tuple T̄ = (T1, . . . , Td) of otherwise arbitrary operators on a Hilbert space, there is an associated Dirac operator DT̄ . Significant attributes of the d-tuple are best expressed in terms of DT̄ , including the Taylor spectrum and the notion of Fredholmness. In fact, all properties of T̄ 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 T̄ which are a) Fredholm and and b) graded, the curvature invariant K(T̄ ) is stable under compact perturbations. We do not know if this stability persists when T̄ is Fredholm but ungraded, though there is concrete evidence that it does. Introduction. We introduce an abstract notion of Dirac operator in complex dimension d = 1, 2, . . . and we show that this theory of Dirac operators actually coincides with the theory of commuting d-tuples of operators on a common Hilbert space H (see Theorem A of section 3). The homology and cohomology of Dirac operators is discussed in general terms, and we relate the homological picture to classical spectral theory by describing its application to concrete problems involving the solution of linear equations of the form T1x1 + T2x2 + · · ·+ Tdxd = y given y and several commuting operators T1, T2, . . . , Td. These developments grew out of an attempt to understand the stability properties of a curvature invariant introduced in a previous paper (see [3], [4]), and to find an appropriate formula that expresses the curvature invariant as the index of some operator. The results are presented in section 4 (see Theorem B and its corollary). While there is a large literature concerning Taylor’s cohomological notion of joint spectrum for commuting sets of operators on a Banach space, less attention has been devoted to the Dirac operator that emerges naturally in the context of Hilbert spaces (however, see sections 4 through 6 of [6], where the operator B+B∗ On appointment as a Miller Research Professor in the Miller Institute for Basic Research in Science. Support is also acknowledged from NSF grant DMS-9802474