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2000 | 143 | 2 | 121-144
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Discrete Wiener-Hopf operators on spaces with Muckenhoupt weight

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
143
Numer
2
Strony
121-144
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-10-14
Twórcy
autor
  • Fakultät für Mathematik, TU Chemnitz, D-09107 Chemnitz, Germany
autor
  • Fakultät für Mathematik, TU Chemnitz, D-09107 Chemnitz, Germany
Bibliografia
  • [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.
  • [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.
  • [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).
  • [8] R. V. Duduchava, On discrete Wiener-Hopf equations, ibid., 42-59 (in Russian).
  • [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.
  • [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.
  • [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.
  • [15] R. Schneider, Integral equations with piecewise continuous coefficients in the $L^p$ spaces with weight, J. Integral Equations 9 (1985), 135-152.
  • [16] I. B. Simonenko, Some general questions of the theory of the Riemann boundary value problem, Math. USSR-Izv. 2 (1968), 1091-1099.
  • [17] I. Spitkovsky, Singular integral operators with PC symbols on the spaces with general weights, J. Funct. Anal. 105 (1992), 129-143.
  • [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.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv143i2p121bwm
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