AI 中文总结
该论文在非交换环面上构建椭圆型拟微分算子的复幂、对数和扇形投影理论，基于前篇演算，确定相关性质、公式并建立增长界，得到相关结果类似物，为谱几何应用奠定解析基础。
AI 中文摘要
本文发展了非交换环面上椭圆型拟微分算子的复幂、对数和扇形投影的系统理论，将Seeley等人的经典构造扩展到该情形。基于前篇的参数型拟微分演算，构建与给定射线相关的复幂，证明其构成具有半群性质的拟微分算子全纯族并计算其符号。通过新的拟微分族全纯演算在算子、Schatten和迹类拓扑中建立垂直带的指数增长界。确定对数为拟微分算子并导出相关迹公式。将扇形投影构造为围道积分并证明其为零阶。得到Wodzicki、Okikiolu和Gaarde - Grubb结果的非交换环面类似物，为后续论文中的谱几何应用提供解析基础。
英文摘要
This paper develops a systematic theory of complex powers, logarithms, and sectorial projections of elliptic pseudodifferential operators on noncommutative tori, extending to this setting the classical constructions of Seeley and others. Building on the parametric pseudodifferential calculus of the prequel~\cite{LP:Part1}, we construct the complex powers associated with a given ray, show that they form a holomorphic family of pseudodifferential operators with the semigroup property, and compute their symbols. We further establish exponential growth bounds on vertical strips in the operator, Schatten, and trace-class topologies by means of a new holomorphic calculus for pseudodifferential families. The logarithm is identified as a pseudodifferential operator whose symbol is determined by the resolvent symbol, and the associated trace formula is derived. Sectorial projections are constructed as contour integrals and shown to be of order zero. This yields analogues for noncommutative tori of results of Wodzicki, Okikiolu, and Gaarde--Grubb, and provides the analytic foundations for the spectral-geometric applications developed in subsequent papers.
Comments61 pages
