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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.

Related project(s):
80Nonlocal boundary problems: Index theory and semiclassical asymptotics

We study the Fredholm solvability for a new class of nonlocal boundary value problems associated with group actions on smooth manifolds. Namely, we consider the case in which the group action is defined on an ambient manifold without boundary and does not preserve the manifold with boundary on which the problem is stated. In particular, the group action does not map the boundary to itself. The orbits of the boundary under the group action split the manifold into subdomains, and this decomposition, being combined with the C*-algebra techniques, plays an important role in our approach to the analysis of the problem.

Related project(s):
80Nonlocal boundary problems: Index theory and semiclassical asymptotics