The Coburn-Simonenko theorem for some classes of Wiener-Hopf plus Hankel operators

Abstract

Wiener-Hopf plus Hankel operators W(a)+ H(b) : Lp(R+) ? Lp(R+) with generating functions a and b from a subalgebra of L?(R) containing almost periodic functions and Fourier images of L1(R)-functions are studied. For a and b satisfying the so-called matching condition a(t)a(?t) = b(t)b(?t), t ? R, we single out some classes of operators W(a)+ H(b) which are subject to the Coburn-Simonenko theorem.

Generalized inverses and solution of equations with Toeplitz plus Hankel operators

“…However, at the moment there is no description for the kernels of such operators. Nevertheless, an investigation of Wiener-Hopf plus Hankel operators in the situation mentioned, has been started in [4]. As soon as that work will be completed, relevant results can be also used for solution of non-homogeneous Wiener-Hopf plus Hankel equations.…”

Generalized Toeplitz Plus Hankel Operators: Kernel Structure and Defect Numbers

“…In Section 3, some classes of generalized Toeplitz plus Hankel operators, where Coburn-Simonenko theorem holds, are studied. For classical Toeplitz plus Hankel operators similar problems are discussed in [3,7,8], whereas [6] deals with Wiener-Hopf plus Hankel operators. In Section 4, a decomposition for the kernels of special scalar Toeplitz operators is derived.…”

Wiener-Hopf plus Hankel operators: Invertibility Problems

“…The factorization (7) has been used in the construction of one-sided inverses for the Wiener-Hopf operators W (g).…”

“…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).…”