The Dirac operator of a commuting d-tuple
| dc.creator | Arveson, William | |
| dc.date | 2000-05-31 | |
| dc.date | 2000-06-11 | |
| dc.date.accessioned | 2026-07-07T04:35:36Z | |
| dc.date.available | 2026-07-07T04:35:36Z | |
| dc.description | 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. | |
| dc.description | 20 pages. Revision includes some examples and explicit computations. There are no other substantive changes | |
| dc.identifier | https://arxiv.org/abs/math/0005285 | |
| dc.identifier | http://arxiv.org/abs/math/0005285 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/59308 | |
| dc.subject | Operator Algebras | |
| dc.title | The Dirac operator of a commuting d-tuple | |
| dc.type | text |