# Discrete Wiener-Hopf operators on spaces with Muckenhoupt weight

Studia Mathematica (2000)

• Volume: 143, Issue: 2, page 121-144
• ISSN: 0039-3223

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## Abstract

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The discrete Wiener-Hopf operator generated by a function $a\left({e}^{i\theta }\right)$ with the Fourier series ${\sum }_{n\in ℤ}{a}_{n}{e}^{in\theta }$ is the operator T(a) induced by the Toeplitz matrix ${\left({a}_{j-k}\right)}_{j,k=0}^{\infty }$ on some weighted sequence space ${l}^{p}\left({ℤ}_{+},w\right)$. We assume that w satisfies the Muckenhoupt ${A}_{p}$ condition and that a is a piecewise continuous function subject to some natural multiplier condition. The last condition is in particular satisfied if a is of bounded variation. Our main result is a Fredholm criterion and an index formula for T(a). It implies that the essential spectrum of T(a) results from the essential range of a by filling in certain horns between the endpoints of each jump. The shape of these horns is determined by the indices of powerlikeness of the weight w.

## How to cite

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Böttcher, A., and Seybold, M.. "Discrete Wiener-Hopf operators on spaces with Muckenhoupt weight." Studia Mathematica 143.2 (2000): 121-144. <http://eudml.org/doc/216812>.

@article{Böttcher2000,
abstract = {The discrete Wiener-Hopf operator generated by a function $a(e^\{iθ\})$ with the Fourier series $∑_\{n∈ℤ\} a_n e^\{inθ\}$ is the operator T(a) induced by the Toeplitz matrix $(a_\{j-k\})_\{j,k = 0\}^∞$ on some weighted sequence space $l^p(ℤ_\{+\}, w)$. We assume that w satisfies the Muckenhoupt $A_p$ condition and that a is a piecewise continuous function subject to some natural multiplier condition. The last condition is in particular satisfied if a is of bounded variation. Our main result is a Fredholm criterion and an index formula for T(a). It implies that the essential spectrum of T(a) results from the essential range of a by filling in certain horns between the endpoints of each jump. The shape of these horns is determined by the indices of powerlikeness of the weight w.},
author = {Böttcher, A., Seybold, M.},
journal = {Studia Mathematica},
keywords = {discrete Wiener-Hopf operators; weighted spaces of sequences; Muckenhoupt weight; Fredholmness; essential spectrum},
language = {eng},
number = {2},
pages = {121-144},
title = {Discrete Wiener-Hopf operators on spaces with Muckenhoupt weight},
url = {http://eudml.org/doc/216812},
volume = {143},
year = {2000},
}

TY - JOUR
AU - Böttcher, A.
AU - Seybold, M.
TI - Discrete Wiener-Hopf operators on spaces with Muckenhoupt weight
JO - Studia Mathematica
PY - 2000
VL - 143
IS - 2
SP - 121
EP - 144
AB - The discrete Wiener-Hopf operator generated by a function $a(e^{iθ})$ with the Fourier series $∑_{n∈ℤ} a_n e^{inθ}$ is the operator T(a) induced by the Toeplitz matrix $(a_{j-k})_{j,k = 0}^∞$ on some weighted sequence space $l^p(ℤ_{+}, w)$. We assume that w satisfies the Muckenhoupt $A_p$ condition and that a is a piecewise continuous function subject to some natural multiplier condition. The last condition is in particular satisfied if a is of bounded variation. Our main result is a Fredholm criterion and an index formula for T(a). It implies that the essential spectrum of T(a) results from the essential range of a by filling in certain horns between the endpoints of each jump. The shape of these horns is determined by the indices of powerlikeness of the weight w.
LA - eng
KW - discrete Wiener-Hopf operators; weighted spaces of sequences; Muckenhoupt weight; Fredholmness; essential spectrum
UR - http://eudml.org/doc/216812
ER -

## References

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