On the index of nonlocal elliptic operators for the group of dilations

This paper considers the index problem for nonlocal elliptic operators associated with actions of discrete groups. The situation in which the action is isometric was considered in the general case (even for infinite groups) in book [1]. In this paper, we consider the situation of a nonisometric action. This situation is much more complicated, and we study it for the example of the group of dilations acting on the sphere of any dimension. The method for studying the problem consists in a realization of a (scalar) nonlocal operator as an operator acting on the sections of infinite dimensional bundles on the orbit space of the group action. For the operator thus obtained, we introduce the notion of ellipticity, prove a finiteness theorem, and give an index formula. © 2010 Pleiades Publishing, Ltd.