Functional Calculus for Elliptic Operators on Noncommutative Tori, I

TL;DR

This paper develops a new parametric pseudodifferential calculus for noncommutative tori, enabling the analysis of elliptic operators and their complex powers within this framework, extending previous methods.

Contribution

It introduces a natural parametric calculus on noncommutative tori that includes resolvents of elliptic operators, not limited to differential operators, and proves complex powers are pseudodifferential.

Findings

Resolved the calculus contains resolvents of non-differential elliptic operators.

Proved complex powers of positive elliptic operators are pseudodifferential.

Confirmed a conjecture by Fathi-Ghorbanpour-Khalkhali.

Abstract

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-Khalkhali.