On the Fredholm and Weyl spectrum of several commuting operators

The Fredholm index of a pair of commuting operators

This paper concerns Fredholm theory in several variables, and its applications to Hilbert spaces of analytic functions. One feature is the introduction of ideas from commutative algebra to operator theory. Specifically, we introduce a method to calculate the Fredholm index of a pair of commuting operators. To achieve this, we define and study the Hilbert space analogs of Samuel multiplicities in commutative algebra. Then the theory is applied to the symmetric Fock space. In particular, our results imply a satisfactory answer to Arveson's program on developing a Fredholm theory for pure $d$-con…

The Dirac operator of a commuting d-tuple

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, an…

几乎交换的自伴算子与测量

Almost commuting self-adjoint operators and measurements

本文研究无限维可分希尔伯特空间中几乎交换自伴算子何时可逼近为交换自伴算子，利用合成谱与Fredholm指标给出n=2时的充要条件，并说明n≥3时结论一般不成立，回应了Mumford关于量子测量的问题。

Functional calculus and joint torsion of pairs of almost commuting operators

This paper investigates the transformation of determinants of pairs of Fredholm operators with trace class commutators. We study the extent to which the functional calculus commutes, modulo operator ideals, with projections in a finitely summable Fredholm module. As an application, we recover in particular some results of R. Carey and J. Pincus on determinants and Tate tame symbols. Additionally, we obtain variational formulas for joint torsion.

On Drazin invertible C*-operators and generalized C*-Weyl operators

In this paper, we obtain several extensions of semi-Fredholm theory on Hilbert modules by generalizing in this setting their classical counterparts regarding Weyl operators and Drazin invertible operators.

Joint torsion equals the determinant invariant

A determinant in algebraic $K$-theory is associated to any two almost commuting Fredholm operators. On the other hand, one can calculate a homologically defined invariant known as joint torsion. We answer in the affirmative a conjecture of Richard Carey and Joel Pincus, namely that these two invariants agree. In particular, this implies that joint torsion is norm continuous, depends only on the images of the operators modulo trace class, and satisfies the expected Steinberg relations. Moreover, we show that the determinant invariant of two commuting operators can be computed simply as a determ…