The Dirac Operator of a Commuting d-Tuple
By W. Arveson. 2000
98 citations recorded by the source corpus.
Abstract Given a commuting d-tuple T=(T1, …, Td) of otherwise arbitrary operators on a Hilbert space, there is an associated Dirac operator DT. Significant attributes of the d-tuple are best expressed in terms of DT, 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 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 T that 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, although there is concrete evidence that it does.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/7603a9dc228d29d7a9aa1f45eac799ed7bfefdf9
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In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- p-Summable Commutators in Dimension d (2003, methodology): “ If one replaces e 1,...,ed with a different orthonormal basis for Z, then of course one changes D; but the two Dirac operators are naturally isomorphic in a sense that we will not spell out here (see [Arv02]). Thus the Dirac operator of T¯ is uniquely determined by T¯ up to isomorphism. The multiope”
- Quotients of standard Hilbert modules (2005, unknown): “The cohomology of the Koszul complex is related to the Dirac operator as follows: B ( H ⊗ Λ k E ) has finite codimension in ker( B H ⊗ Λ k +1 E ), for every k = 0 , 1 , . . . , d iff D is a self-adjoint Fredholm operator [Arv02].”
- Essentially Reductive Hilbert Modules (2004, background): “In other cases, one must resort to different measures [9], [3].”
- Essentially normal Hilbert modules and K-homology (2008, methodology): “Arveson raised a conjecture about the essentially normality or p-essentially normality (p > d) of graded submodules of H2 d ⊗ Cr with finite multiplicity r [3–7].”
- Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients (2013, background): “This problem was taken up in [22] for graded Hilbert modules.”
- m-Isometric Commuting Tuples of Operators on a Hilbert Space (2006, background): “A more recent development along these lines was first introduced by Drury in [13] and was further developed and popularized by Arveson in a series of papers, [4], [5], and [6].”
- A new kind of index theorem (2005, methodology): “Then we discuss a conjecture of Arveson [4] on homogeneous submodules of the m-shift Hilbert space H(2) m and propose some refinements of it.”
- Essentially Reductive Hilbert Modules II (2006, background): “0 Introduction In his study [1] of the m-shift Hilbert module H(2) m over C[z ], the polynomial algebra in m variables, Arveson formulated a provocative conjecture which has attracted the attention of several researchers.”
- On the index of invariant subspaces in spaces of analytic functions of several complex variables (2005, background): “As was pointed out in [11], it follows from a result of Curto ([16]) that ðMz lÞ jM is Fredholm for all l A Bd whenever MzjM is an essentially normal tuple (i.”