This paper is devoted to the application of the \(C^*\)-algebraic method to the study of the Fredholm property for some classes of \(G\)-pseudodifferential operators with piecewise continuous coefficients and \(G\)-boundary value problems. The typical model operators of this sort are the operators of the form \(\sum(A_jTg_j)\), where \(A_j\) are pseudodifferential operators on a manifold \(M\) and \((Tg)(x)=f(\alpha^{-1}g(x))\), where \(\{\alpha g\}_{g\in G}\) is a representation of a group \(G\) on \(M\). The main purpose of this paper is to construct the symbolic calculus for this class of non-local operators, to prove that a \(G\)-pseudodifferential operator on a compact manifold is Fredholm in the corresponding Sobolev spaces if and only if its principal symbol is invertible, and to prove the index theorem.
The authors present the main ideas, methods and some results of the theory; for a detailed discussion of the question considered here, they refer to their books [Functional-differential equations. I: \(C^*\)-theory. Harlow: Longman (1994; Zbl 0799.34001); Functional differential equations. II: \(C^*\)-applications. Part 1: Equations with continuous coefficients. Harlow: Longman (1998; Zbl 0936.35207); Part 2: Equations with discontinuous coefficients and boundary value problems. Harlow: Longman (1998; Zbl 0936.35208)].
For the entire collection see [Zbl 0942.00023].