Geodesic
On~integro-functional operators with a~shift which is not one-to-one
Izvestiya. Mathematics , Tome 19 (1982) no. 3, pp. 479-493.

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Functional and singular integro-functional operators with nonunivalent shift are considered. The spectrum of a weighted shift operator in the space $L_p(\Gamma)$, $1\leqslant p\leqslant\infty$, is studied in the case where the shift is an expanding nonunivalent mapping of a smooth finite-dimensional mapping of a manifold $\Gamma$. Necessary and sufficient conditions for the Fredholm property are obtained, as well as a formula for computing the index of a singular integral operator with nonunivalent expanding shift of the unit circle. Bibliography: 17 titles. 
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Yu. D. Latushkin. On~integro-functional operators with a~shift which is not one-to-one. Izvestiya. Mathematics , Tome 19 (1982) no. 3, pp. 479-493. http://geodesic.mathdoc.fr/item/IM2_1982_19_3_a1/

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