## 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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