The Coburn-Simonenko theorem for some classes of Wiener-Hopf plus Hankel operators (Q2815257)

This clearly written paper starts with a review of results on Toeplitz plus Hankel operators \(T(a)+H(b)\), remarking that Wiener-Hopf plus Hankel operators have received less attention in the literature. The main motivation of the paper is to establish the Coburn-Simonenko theorem for such operators following the footsteps of the author's results obtained in [\textit{V. D. Didenko} and \textit{B. Silbermann}, Integral Equations Oper. Theory Vol. 80, No. 1, 1--31 (2014; Zbl 1314.47041)].NEWLINENEWLINE The generating functions belong to the space defined as direct sum of almost periodic functions with absolutely convergent Fourier series and Fourier images of \(L^1(\mathbb R)\) functions. As a preparation for the main result, the authors establish certain relations between the kernels of Wiener-Hopf plus Hankel operators and matrix Wiener-Hopf operators in Section 2. In particular, they introduce the notion of a matching pair \((a,b)\), its subordinated pair and matching function, cf. reference above.NEWLINENEWLINE The main result is a variant of the Coburn-Simonenko theorem for operators \(W(a)\pm H (a)\), \(W(a)-H(a\chi)\), and \( W(a)+H (a\chi^{-1})\), when \(a\) is invertible and \(\chi(t)=(t-i)/(t+i)\), \(t\in\mathbb R\), which states that their kernel or cokernel is trivial space. In addition, possible generalizations of the main theorem are discussed and some of them are proved.