Invertibility of Toeplitz + Hankel Operators and Singular Integral Operators with Flip - The case of smooth generating functions

Abstract

It is well known that a Toeplitz operator is invertible if and only if its symbols admits a canonical Wiener-Hopf factorization, where the factors satisfy certain conditions. A similar result holds also for singular integral operators. More general, the dimension of the kernel and cokernel of Toeplitz or singular integral operators which are Fredholm can be expressed in terms of the partial indices of an associated Wiener-Hopf factorization problem. In this paper we establish corresponding results for Toeplitz + Hankel operators and singular integral operators with flip under the assumption that the generating functions are sufficiently smooth. We are led to a slightly different factorization problem, in which so-called characteristic pairs instead of the partial indices appear. These pairs provide the relevant information about the dimension of the kernel and cokernel and thus answer the invertibility problem.