The index of elliptic operators on compact manifolds

Introduction.In his paper [ló] Gel'fand posed the general problem of investigating the relationship between topological and analytical invariants of elliptic differential operators.In particular he suggested that it should be possible to express the index of an elliptic operator (see §1 for the definition) in topological terms.This problem has been taken up by Agranovic [2; 3], Dynin [3; 14; 15], Seeley [20;21 ] and Vol'pert [22 ] who have solved it in special cases.The purpose of this paper is to give a general formula for the index of an elliptic operator on any compact oriented differentiable manifold (Theorem 1).As a special case of this formula we get the Hirzebruch-Riemann-Roch theorem for any compact complex manifold (Theorem 3).This was previously known only for projective algebraic manifolds.Some other special cases, of interest in differential topology, are discussed in §3.We are greatly indebted to A. P. Calderon, L. Nirenberg, and R. T. Seeley for their generous help. Elliptic operators.Let X be a compact oriented smooth mani-