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Journal of Operator Theory

Volume 85, Issue 1, Winter 2021  pp. 229-256.

Fredholm conditions for invariant operators: finite abelian groups and boundary value problems

Authors: Alexandre Baldare (1), Remi Come (2), Matthias Lesch (3), Victor Nistor (4)
Author institution: (1) Universite Lorraine, 57000 Metz, France
(2) Universite Lorraine, 57000 Metz, France
(3) Mathematisches Institut, Univ. Bonn, Endenicher Allee 60, 53115 Bonn, Germany
(4) Universite Lorraine, 57000 Metz, France

Summary: Let $\Gamma$ be a finite abelian group acting on a smooth, compact manifold $M$ without boundary and let $P \in \psi^m(M; E_0, E_1)$ be a $\Gamma$-invariant, classical, pseudodifferential operator acting between sections of two $\Gamma$-equivariant vector bundles. Let $\alpha$ be an irreducible representation of $\Gamma$. We obtain necessary and sufficient conditions for the restriction $\pi_\alpha(P) : H^s(M; E_0)_\alpha \to H^{s-m}(M; E_1)_\alpha$ of $P$ between the $\alpha$-isotypical components of Sobolev spaces to be Fredholm.

Keywords: Fredholm operator, pseudodifferential operator, finite abelian group, invariant operator, isotypical component, elliptic theory

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