Traces of \(G\)-operators concentrated on submanifolds
The paper concerns the problem of the definition of the trace of an operator, namely given a manifold \(M\), a submanifold \(X\) and an operator \(P\) on \(M\), one aims to find an operation assigning to \(P\) an operator \(P|_X\), see for example \textit{S. P. Novikov} and \textit{B. Yu. Sternin} [Sov. Math., Dokl. 7, 1373--1376 (1966; Zbl 0174.41302); translation from Dokl. Akad. Nauk SSSR 170, 1265--1268 (1966)]. Here, the author considers the case of a discrete group \(G\) on the manifold \(M\) and operators on \(M\) of the form \[ P= \sum_{g\in G}P_g T_g, \] where \(P_g\) is a sequence of pseudodifferential operators on \(M\) of a fixed order \(m\) and \(T_g\) is the shift operator \(T_gu(x)= u(g^{-1}x)\) induced by an element \(g\in G\). Under a condition on the order \(m\), the author proves that the trace on a submanifold \(X\) is well defined and give for \(P|_X\) a precise expression.
scientific article; zbMATH DE number 6909061
scientific article