Fredholm Criteria for $G$-pseudodifferential Operators

TL;DR

This paper establishes new criteria for when G-pseudodifferential operators are Fredholm, based on group actions, Simonenko principles, and symbol invertibility, enhancing understanding of their analytical properties.

Contribution

It introduces a novel Fredholm criterion for G-pseudodifferential operators on manifolds with group actions, extending previous results to a broader class of operators.

Findings

Derived a Fredholm criterion using Simonenko principle for G-pseudodifferential operators.

Characterized Fredholm property via invertibility of symbols when G is finite.

Extended criteria to operators acting on Sobolev spaces of sections over manifolds.

Abstract

Let $G$ be a compact Lie group that acts smoothly on a closed manifold $M$. Using a general Simonenko principle, we derive a novel criterion for the Fredholm property of $G$-pseudodifferential operators acting on Sobolev spaces of sections of vector bundles over $M$. In case the group is finite, we obtain a further characterization of the Fredholm property of $G$-pseudodifferential operators in terms of the invertibility of suitable symbols.