On Graded K-theory, Elliptic Operators and the Functional Calculus

分次李群作用的 Shubin 演算

Shubin calculi for actions of graded Lie groups

本文用群胚方法为分次李群在分次向量空间上的作用建立渐近完备的 Shubin 型伪微分算子演算，刻画椭圆性、Rockland 条件、映射性质与谱。

Functional Calculus for Elliptic Operators on Noncommutative Tori, I

In this paper, we introduce a parametric pseudodifferential calculus on noncommutative $n$-tori which is a natural nest for resolvents of elliptic pseudodifferential operators. Unlike in some previous approaches to parametric pseudodifferential calculi, our parametric pseudodifferential calculus contains resolvents of elliptic pseudodifferential operators that need not be differential operators. As an application we show that complex powers of positive elliptic pseudodifferential operators on noncommutative $n$-tori are pseudodifferential operators. This confirms a claim of Fathi-Ghorbanpour-K…

The regularity of the $η$ function for the Shubin calculus

We prove the regularity of the $η$ function for classical pseudodifferential operators with Shubin symbols. We recall the construction of complex powers and of the Wodzicki and Kontsevich-Vishik functionals for classical symbols on $\mathbb{R}^{n}$ with these symbols. We then define the $ζ$ and $η$ functions associated to suitable elliptic operators. We compute the $K_{0}$ group of the algebra of zero-order operators and use this knowledge to show that the Wodzicki trace of the idempotents in the algebra vanishes. From this, it follows that the $η$ function is regular at 0 for any self-adjoint…

On elliptic differential operators with shifts: II. The cohomological index formula

This paper is a continuation of arXiv:0706.3511, where we obtained a local index formula for matrix elliptic operators with shifts. Here we establish a cohomological index formula of Atiyah-Singer type for elliptic differential operators with shifts acting between section spaces of arbitrary vector bundles. The key step is the construction of closed graded traces on certain differential algebras over the symbol algebra for this class of operators.

过滤微积分与R-作用的交叉积

Filtered Calculus and crossed products by R-actions

本文证明了过滤流形切向群胚C*-代数的核与0阶伪微分算子在自然R-作用下的交叉积同构，并利用分级幂零李群的结构结果推广了主符号代数的分解。

Green Operators in the Edge Calculus

The task to construct parametrices of elliptic differential operators on a manifold with edges requires a calculus of operators with a two-component principal symbolic hierarchy, consisting of (edge-degenerate) interior and (operator-valued) edge symbols. This so-called edge-algebra can be interpreted as a generalisation of the (pseudo-differential) algebra of boundary value problems without the transmission property at the boundary. We study new properties of the edge-algebra, in particular, what concerns the role of Green operators and their kernel representations.