ISSN 0001-4346, Mathematical Notes, 2015, Vol. 97, No. 4, pp. 502–509. © Pleiades Publishing, Ltd., 2015. Original Russian Text © A. A. Arutyunov, 2015, published in Matematicheskie Zametki, 2015, Vol. 97, No. 4, pp. 493–502. Reduction of Nonlocal Pseudodifferential Operators on a Noncompact Manifold to Classical Pseudodifferential Operators on a Double-Dimensional Compact Manifold A. A. Arutyunov * Moscow Institute of Physics and Technology, Dolgoprudnyi, Moscow Region, Russia Received February 17, 2014 Abstract—In the present paper, we obtain some Fredholmness criteria and a formula for the index of nonlocal pseudodifferential operators generated by shifts and multiplications by periodic functions and acting in the Schwartz space S (R n ). These results significantly differ from the results known to the author in this field of study, namely, the Fredholmness criteria and the formula for the index of nonlocal pseudodifferential operators acting over the noncompact manifold R n are obtained here for the first time. DOI: 10.1134/S0001434615030220 Keywords: nonlocal pseudodifferential operator, Fredholmness criterion, index of elliptic pseudodifferential operators, Schwartz space. 1. INTRODUCTION The problem of calculating the index of elliptic pseudodifferential operators was first posed by Gelfand in 1969 in [1]. In 1962, Atiyah and Singer published their well-known paper [2] with the Atiyah–Singer formula which permits calculating the index of an elliptic pseudodifferential operator on a compact manifold in terms of homotopy invariants. One of the main directions of development in studying the index of elliptic operators has to do with nonlocal pseudodifferential operators (PDO with shifts). As in the case of ordinary pseudodifferential operators, there are many rather general works, where the index is calculated for nonlocal pseudodif- ferential operators on compact manifolds. Thus, a formula for calculating the index of nonlocal PDO’s with a finite group of shifts is given in [3]. For infinite groups, we mention the monograph [4], where this problem is solved in the case of isometric actions of the group, i.e., when the metric on the manifold is preserved. In the present paper, we obtain some Fredholmness criteria and a formula for the index of nonlocal pseudodifferential operators acting in the Schwartz space S (R n ) and generated by shifts and multiplica- tions by periodic functions. These results significantly differ from the above-listed results in that they are the first in which the Fredholmness criteria and the formula for the index of nonlocal pseudodifferential operators acting over the noncompact manifold R n were obtained. We need the results obtained earlier in [5] and [6] by the author and Professor A. S. Mishchenko. Let us recall the basic results obtained in these papers. We define the class of symbols of pseudodifferential operators acting in the Schwartz space S (R n ) which is equivalent to the class of so-called SG-operators (see [7], [8]). The class of symbols which are interesting for us consists of infinitely differentiable functions σ(x, ξ ) satisfying the inequality |∂ α x ∂ β ξ σ(x, ξ )|≤ C α,β (1 + |x|) m 1 −|α| (1 + |ξ |) m 2 −|β| for all x, ξ (1.1) * E-mail: Andronick.Arutyunov@gmail.com 502