Operator theory
Fredholm conditions for operators invariant with respect to compact Lie group actions
Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1135-1143.

Let $G$ be a compact Lie group acting smoothly on a smooth, compact manifold $M$, let $P\in {\psi }^{m}\left(M;{E}_{0},{E}_{1}\right)$ be a $G$–invariant, classical pseudodifferential operator acting between sections of two $G$-equivariant vector bundles ${E}_{i}\to M$, $i=0,1$, and let $\alpha$ be an irreducible representation of the group $G$. Then $P$ induces a map ${\pi }_{\alpha }\left(P\right):{H}^{s}{\left(M;{E}_{0}\right)}_{\alpha }\to {H}^{s-m}{\left(M;{E}_{1}\right)}_{\alpha }$ between the $\alpha$-isotypical components. We prove that the map ${\pi }_{\alpha }\left(P\right)$ is Fredholm if, and only if, $P$ is transversally $\alpha$-elliptic, a condition defined in terms of the principal symbol of $P$ and the action of $G$ on the vector bundles ${E}_{i}$.

Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.257
Classification: 47A53,  58J40,  57S15,  47L80,  46N20
Alexandre Baldare 1; Rémi Côme 2; Victor Nistor 2

1 Institut fur Analysis, Welfengarten 1, 30167 Hannover, Germany
2 Université Lorraine, 57000 Metz, France
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Alexandre Baldare; Rémi Côme; Victor Nistor. Fredholm conditions for operators invariant with respect to compact Lie group actions. Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1135-1143. doi : 10.5802/crmath.257. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.257/

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