On the symbol of nonlocal operators in Sobolev spaces (Q647721)

Let us consider a diffeomorphism \(g: M\to M\) of a smooth closed manifold an denote \(T^k\), \(k\in\mathbb{Z}\), the powers of the operator \(Tu(x)= u(g(x))\). The author studies operators \(D\) in the form of a finite sum, acting on Sobolev spaces: \[ D= \sum_k D_k T^k: H^s(M)\to H^{s-m}(M), \] where \(D_k\) are classical pseudodifferential operators of order \(\leq m\) on the manifold \(M\). Operators of this form were considered, for example, \textit{A. B. Antonevich} [Math. USSR, Izv. 34, No. 1, 1--21 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 1, 3--24 (1989; Zbl 0798.35045)]. They have not pseudo-local character and can be regarded as Fourier integral operators on \(M\). The author defines here the notion of symbol \(\sigma(D)\), as vector-valued function on \(T^*M/\{0\}\). The composition formula \(\sigma(DD')= \sigma(D)\sigma(D')\) is proved and Fredholm properties are deduced. The relevant case when \(M\) is given by the sphere \(\mathbb{S}^n\) is studied in detail and explicit conditions for Fredholmness are obtained.