Uniformization and an Index Theorem for Elliptic Operators Associated with Diffeomorphisms of a Manifold

Abstract

We consider the index problem for a wide class of nonlocal elliptic operators on a smooth closed manifold, namely differential operators with shifts induced by the action of an isometric diffeomorphism. The key to the solution is the method of uniformization: We assign to the nonlocal problem a pseudodifferential operator with the same index, acting in sections of an infinite-dimensional vector bundle on a compact manifold. We then determine the index in terms of topological invariants of the symbol, using the Atiyah-Singer index theorem.

16 pages, no figures

Cited by in corpus (5)

• Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations
• Elliptic theory for operators associated with diffeomorphisms of smooth manifolds
• Elliptic Operators Associated with Groups of Quantized Canonical Transformations
• Equivariant Algebraic Index Theorem
• $C^{\ast }$-algebras of transmission problems and elliptic boundary value problems with shift operators