ar X iv : m at h / 05 11 12 7 v 1 [ m at h . FA ] 5 N ov 2 00 5 Factorization theory for Wiener - Hopf plus Hankel operators with almost periodic symbols

A factorization theory is proposed for Wiener-Hopf plus Hankel operators with almost periodic Fourier symbols. We introduce a factorization concept for the almost periodic Fourier symbols such that the properties of the factors will allow corresponding operator factorizations. Conditions for left, right, or both-sided invertibility of the Wiener-Hopf plus Hankel operators are therefore obtained upon certain indices of the factorizations. Under such conditions, the one-sided and two-sided inverses of the operators in study are also obtained. 1. Historical background We would like to begin with a historical review about Hankel operators, including Toeplitz operators and Wiener-Hopf operators, till Wiener-Hopf plus Hankel operators. The story of Hankel operators starts with Hermann Hankel and with his Ph.D. thesis [13] published in 1861. Here Hankel studied determinants of infinite complex matrices with entries defined by ajk = aj+k (j, k ≥ 0), where a = {aj}j≥0 is a sequence of complex numbers. These matrices are the therefore called Hankel matrices and have the special form HM =   a0 a1 a2 a3 · · · a1 a2 a3 a4 · · · a2 a3 a4 a5 · · · a3 a4 a5 a6 · · · .. .. .. .. . . .   . In 1881, Kronecker [14] presented a theorem that describes the Hankel matrices of finite rank as the ones that have corresponding power series,