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Let $Gamma$ be a compact group acting on a smooth, compact manifold $M$, let $P in psi^m(M; E_0, E_1)$ be a $Gamma$-invariant, classical pseudodifferential operator acting between sections of two equivariant vector bundles $E_i to M$, $i = 0,1$, and let $alpha$ be an irreducible representation of the group $Gamma$. Then $P$ induces a map $pi_alpha(P) : H^s(M; E_0)_alpha to H^{s-m}(M; E_1)_alpha$ between the $alpha$-isotypical components of the corresponding Sobolev spaces of sections. When $Gamma$ is finite, we explicitly characterize the operators $P$ for which the map $pi_alpha(P)$ is Fredholm in terms of the principal symbol of $P$ and the action of $Gamma$ on the vector bundles $E_i$. When $Gamma = {1}$, that is, when there is no group, our result extends the classical characterization of Fredholm (pseudo)differential operators on compact manifolds. The proof is based on a careful study of the symbol $C^*$-algebra and of the topology of its primitive ideal spectrum. We also obtain several results on the structure of the norm closure of the algebra of invariant pseudodifferential operators and their relation to induced representations. Whenever our results also hold for non-discrete groups, we prove them in this greater generality. As an illustration of the generality of our results, we provide some applications to Hodge theory and to index theory of singular quotient spaces.
We answer the question of when an invariant pseudodifferential operator is Fredholm on a fixed, given isotypical component. More precisely, let $Gamma$ be a compact group acting on a smooth, compact, manifold $M$ without boundary and let $P in psi^m(
In this paper, we derive, from a general Simonenkos local principle, Fredholm criteria for restriction to isotypical components. More precisely, we gave a full proof, of the equivariant local principle for restriction to isotypical components of inva
Let $G$ be a compact Lie group acting smoothly on a smooth, compact manifold $M$, let $P in psi^m(M; E_0, E_1)$ be a $G$--invariant, classical pseudodifferential operator acting between sections of two vector bundles $E_i to M$, $i = 0,1$, and let $a
A C*algebra A generated by a class of zero-order classical pseudodifferential operator on a cylinder RxB, where B is a compact riemannian manifold, containing operators with periodic symbols, is considered. A description of the K-theory index map ass
In cite{APSIII} Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimensio