Kernels of Wiener–Hopf plus Hankel Operators with Matching Generating Functions

Abstract

Considered are Wiener-Hopf plus Hankel operatorswith generating functions a and b from a subalgebra of L ∞ (R) containing almost periodic functions and Fourier images of L 1 (R)-functions. If the generating functions a and b satisfy the matching condition a(t)a(−t) = b(t)b(−t), t ∈ R, an explicit description for the kernels and cokernels of the operators mentioned is given.

Wiener-Hopf plus Hankel operators: Invertibility Problems

“…The next proposition comprises results from [7,12]. For the reader's convenience, they are reformulated in a form suitable for subsequent presentation.…”

“…Remark 2.3 If ν < 0, the corresponding spaces im P ± g are also described in [12]. However, these representations are not used in what follows so that they are not included to the above proposition.…”

“…Let us briefly discuss the case where ν 1 > 0 and ν 2 < 0. As was mentioned in [12], in this situation it is not clear whether the corresponding Wiener-Hopf operator is even normally solvable. Nevertheless, the kernel and cokernel of W (a) + H(b) can still be described.…”

“…Remark 3.4 Although the description of the spaces im P + d and im P + c is available [12], the verification of the conditions (21)-( 22) is not trivial. It depends on the properties of Wiener-Hopf operators constituting the operator ϕ + and may require a lot of effort.…”

“…Thus if a, b ∈ G, then the Coburn-Simonenko Theorem for some classes of operators W(a, b) is established [7], and an efficient description of the space ker W(a, b) is obtained [12]. The aim of this work is to find conditions for onesided invertibility, invertibility and generalized invertibility of the operators W(a, b) and to provide efficient representations for the corresponding inverses when generating functions a and b satisfy the matching condition (5).…”