Desuspension of splitting elliptic symbols I

This paper provides an algorithm for the conversion of the index of an elliptic first-order differential operator A on the torus Y×S¹ into the index of a canonically associated elliptic pseudo-differential operator Q and Y . It is supposed that Y is a closed smooth manifold and that A splits into /t + B_t , where {B_t} is a family of self-adjoint elliptic operators on Y satisfying the periodicity condition B₁ = g B₀ g^–1 for some unitary automorphism g . Then it will be shown that the operator Q ( the desuspension of A ) can be written down explicity in the form Q = P₊ –gP_– where P₊ are projections onto the space of Cauchy data. In the second part , applications are given for the calculation of the index of the general linear conjugation problem (cutting and pasting of elliptic operators) , and the intimate interrelations between the related procedures of algebraic topology, spectral theory and functional analysis are explained . Generalizations in various directions are indicated.