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In this paper, we derive, from a general Simonenko's local principle,Fredholm criteria for restriction to isotypical components. More precisely, wegive a full proof, of the equivariant local principle for restriction toisotypical components of invariant pseudodifferential operators announced in\cite{BCLN2}. Furthermore, we extend this result by relaxing the hypothesismade in the preceding quoted paper.

A general Simonenko local principle and Fredholm condition for isotypical components