# Discrete Wiener-Hopf operators on spaces with Muckenhoupt weight

Studia Mathematica (2000)

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

## Access Full Article

top## Abstract

top## How to cite

topBöttcher, 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

top- [1] A. Böttcher and Yu. I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators, Progr. Math. 154, Birkhäuser, Basel, 1997.
- [2] A. Böttcher and M. Seybold, Wackelsatz and Stechkin's inequality for discrete Muckenhoupt weights, preprint 99-7, TU Chemnitz, 1999.
- [3] A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, Akademie-Verlag, Berlin, 1989, and Springer, Berlin, 1990. Zbl0689.45009
- [4] A. Böttcher and I. Spitkovsky, Wiener-Hopf integral operators with PC symbols on spaces with Muckenhoupt weight, Rev. Mat. Iberoamericana 9 (1993), 257-279. Zbl0779.45002
- [5] L. A. Coburn, Weyl's theorem for non-normal operators, Michigan Math. J. 13 (1966), 285-286.
- [6] R. V. Duduchava, Discrete Wiener-Hopf equations in ${l}^{p}$ spaces with weight, Soobshch. Akad. Nauk Gruzin. SSR 67 (1972), 17-20 (in Russian).
- [7] R. V. Duduchava, On convolution integral operators with discontinuous symbols, Trudy Tbiliss. Mat. Inst. 50 (1975), 33-41 (in Russian). Zbl0411.47028
- [8] R. V. Duduchava, On discrete Wiener-Hopf equations, ibid., 42-59 (in Russian). Zbl0412.47015
- [9] R. V. Duduchava, Integral Equations in Convolution with Discontinuous Presymbols, Singular Integral Equations with Fixed Singularities, and Their Applications to Some Problems of Mechanics, Teubner, Leipzig 1979. Zbl0439.45002
- [10] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud. 116, North-Holland, Amsterdam, 1985.
- [11] I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations, Vols. I and II, Oper. Theory Adv. Appl. 53 and 54, Birkhäuser, Basel, 1992 (Russian original: Shtiintsa, Kishinev, 1973).
- [12] R. Hunt, B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-252. Zbl0262.44004
- [13] N. K. Nikol’skiǐ, On spaces and algebras of Toeplitz matrices acting on ${l}^{p}$, Sibirsk. Mat. Zh. 7 (1966), 146-158 (in Russian).
- [14] S. Roch and B. Silbermann, Algebras of convolution operators and their image in the Calkin algebra, report R-Math-05/90, Karl-Weierstrass-Inst. f. Math., Berlin, 1990. Zbl0717.47019
- [15] R. Schneider, Integral equations with piecewise continuous coefficients in the ${L}^{p}$ spaces with weight, J. Integral Equations 9 (1985), 135-152. Zbl0573.45001
- [16] I. B. Simonenko, Some general questions of the theory of the Riemann boundary value problem, Math. USSR-Izv. 2 (1968), 1091-1099. Zbl0186.13601
- [17] I. Spitkovsky, Singular integral operators with PC symbols on the spaces with general weights, J. Funct. Anal. 105 (1992), 129-143. Zbl0761.45001
- [18] S. B. Stechkin, On bilinear forms, Dokl. Akad. Nauk SSSR 71 (1950), 237-240 (in Russian).
- [19] J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer, Berlin, 1989.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.