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Wiener-Hopf integral operators with \(PC\) symbols on spaces with Muckenhoupt weight. (English) Zbl 0779.45002

Let \(A_ p(p>1)\) denote the set of all nonnegative functions \(w\) on \(\mathbb{R}\) such that the singular integral operator \(S\), \[ (Sf)(x)={1\over\pi i}\int^ \infty_{-\infty}{f(t)\over t-x}dt,\;x\in\mathbb{R} \] is bounded on the space \(L^ p(\mathbb{R},w)\) and let \(W\) be a Wiener-Hopf integral operator defined by the formula \[ (Wf)(x)=\sum^ m_{j=1}{c_ j\over\pi i}\int^ \infty_ 0{e^{i\alpha_ j(t- x)}f(t)\over t-x}dt+\int^ \infty_ 0k(x-t)f(t)dt,\;x>0, \] where \(c_ j\in\mathbb{C}\) and \(\alpha_ j\in\mathbb{R}\) are given numbers and \(k\in L^ 1(\mathbb{R})\) is a given function. The main result of the present paper describes the essential spectrum of \(W\) in the case \(W\) is any weight belonging to \(Ap\). The essential spectrum of \(W\) is the set of all \(\lambda\in\mathbb{C}\) for which \(W-\lambda I\) is not Fredholm.

MSC:

45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45C05 Eigenvalue problems for integral equations
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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