Elliptic translators on manifolds with point singularities (Q1943714)

Consider a smooth closed manifold \(M\) and two closed submanifolds \(Y^p\), \(p= 1,2\), transversally intersecting along a smooth submanifold \(X\), \(\dim X= 0\), \(X\) being the set of singularities of \(Y= Y^1\cup Y^2\). Following the second author, the authors define a pseudodifferential translator \(T_{12}\) acting from \(Y^2\) to \(Y^1\) and develop an elliptic theory for the operators of the form \[ 1+ T: H^s(Y)\to H^s(Y),\quad H^2(Y)= H^{s_1}(Y^1)\oplus H^{s_2}(Y^2). \] The elliptic operators are of Fredholm type (Th. 1) and an index formula for them is found in Th. 2. The index of \((1+T)\) is expressed by the winding number of the invertible on the weight line \(\text{Re\,}z= \alpha\) function \(1+\sigma(T)(z)\), \(\sigma(T)(z)\) being the symbol of corresponding operator family.