We describe the spectrum and the essential spectrum and give an index formula for Wiener-Hopf integral operators with piecewise continuous symbols on the space with a Muckenhoupt weight . Our main result says that the essential spectrum is a set resulting from the essential range of the symbol by joining the two endpoints of each jump by a certain sickle-shaped domain, whose shape is completely determined by the value of and the behavior of the weight at the origin and at infinity.
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Albrecht Böttcher, Ilya Spitkovsky, Wiener-Hopf integral operators with symbols on spaces with Muckenhoupt weight. Rev. Mat. Iberoam. 9 (1993), no. 2, pp. 257–279