Generalized Fredholm properties for invariant pseudodifferential operators

TL;DR

This paper develops a generalized Fredholm theory for invariant pseudodifferential operators on G-bundles, utilizing von Neumann's G-dimension and a generalized Paley-Wiener theorem to establish solvability and index criteria.

Contribution

It introduces a new framework combining G-invariant pseudodifferential operators with von Neumann dimension and Paley-Wiener theorems, extending Fredholm theory.

Findings

Established a generalized L^2 Fredholm theory for invariant operators.

Provided solvability criteria using a generalized Paley-Wiener theorem.

Defined a G-index and transversal dimension with associated Fredholm theory.

Abstract

We define classes of pseudodifferential operators on $G$-bundles with compact base and give a generalized $L^2$ Fredholm theory for invariant operators in these classes in terms of von Neumann's $G$-dimension. We combine this formalism with a generalized Paley-Wiener theorem, valid for bundles with unimodular structure groups, to provide solvability criteria for invariant operators. This formalism also gives a basis for a $G$-index for these operators. We also define and describe a transversal dimension and its corresponding Fredholm theory in terms of anisotropic Sobolev estimates, valid also for similar bundles with nonunimodular structure group.