Functional Calculus on Noncommutative Tori, II. Complex Powers, Logarithms, and Sectorial Projections

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.