Invertibility theory for Toeplitz plus Hankel operators and singular integral operators with flip

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 generally, the dimension of the kernel and cokernel of Toeplitz or singular integral operators which and Fredholm operators can be expressed in terms of the partial indices ϰ 1,…,ϰ N∈ Z of an associated Wiener–Hopf factorization problem. In this paper we establish corresponding results for Toeplitz plus Hankel operators and singular integral operators with flip under the assumption that the generating functions are sufficiently smooth (e.g., Hölder continuous). We are led to a slightly different factorization problem, in which pairs (ϱ 1,ϰ 1),…,(ϱ N,ϰ N)∈{−1,1}× Z , 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.