# Journal of Operator Theory

Volume 62, Issue 1, Summer 2009 pp. 3-31.

A semi-Fredholm theory for Wiener-Hopf-Hankel operators with piecewise almost periodic Fourier symbols**Authors**: L.P. Castro (1) and A.P. Nolasco (2)

**Author institution:**(1) Department of Mathematics and Research Unit "Matematica e Aplicacoes", University of Aveiro, Aveiro, 3810-193 Aveiro, Portugal

(2) Department of Mathematics and Research Unit "Matematica e Aplicacoes", University of Aveiro, Aveiro, 3810-193 Aveiro, Portugal

**Summary:**We present a semi-Fredholm theory for Wiener--Hopf plus/minus Hankel operators acting between $L^2$ Lebesgue spaces, and having piecewise almost periodic Fourier symbols. This means conditions to ensure the Fredholm property and one-sided invertibility of these operators. This is based on some mean values of the representatives at infinity of the Fourier symbols as well as on the discontinuities of certain auxiliary functions. A formula for the sum of the Fredholm indices of these Wiener--Hopf plus and minus Hankel operators is also obtained, and interpreted upon different cases of symmetries of the discontinuities of the Fourier symbols. Several examples are presented, and the (both-sided) invertibility of the operators in study is also discussed.

**Keywords:**Fredholm property, invertibility, Wiener--Hopf--Hankel operators, piecewise almost periodic function.

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