申请者致力于研究函数空间上算子的trace理论。主要工作如下：1.对多圆盘版本Arveson猜测的研究。申请人完全刻画了多圆盘上齐次(拟齐次)商模的本质正规性，研究了对应的p-本质正规性，得到坐标算子之交换子的trace公式。2.对微分算子的trace理论的研究。申请人将研究解析Hilbert模中p-本质正规性问题的思想方法应用到Hamilton系统的特征值问题中，建立了Hill-型公式和trace公式。同时，利用trace公式研究Hamilton系统的线性稳定性，最终将稳定性结论应用在n-体问题中，得到平面三体问题中Lagrange轨道的稳定区域的估计，这是170年以来对Lagrangian轨道稳定区域的首次估计。近年来，申请者在J.Funct.Anal.等杂志发表论文10余篇。本研究拟进一步讨论解析Hilbert模以及微分算子的trace理论，并进一步理解它们之间的内在联系。

The p-essential normality of analytic Hilbert modules plays an important role in the study of BDF theory. The idea to study the Hilbert modules can be used to study the eigenvalue problem of differential equations. In the past decade, we gave the completely characterization of homogenous and quasi-homogenous quotient modules of Hardy module over the polydisc. From 2009, we began to study the Hill-type formula and trace formula for quasi-periodic orbits in Hamiltonian system. Combined with the relative Morse index theory, we give some stability criteria for Hamiltonian systems. Finally, we use the stability criteria to study the stable region and hyperbolic region of Lagrangian orbits in planar 3-body problem. Recently, we publish our papers in J. Funct. Anal., Arch. Ration. Mech. Anal, Math.Z. ect. In this project, we wish to continue the studying of trace theory of analytic Hilbert modules, the differential operators, and learn the.essential relationship between them.