Discrete Wiener-Hopf operators on spaces with Muckenhoupt weight

Tom 143 / 2000

A. Böttcher, M. Seybold Studia Mathematica 143 (2000), 121-144 DOI: 10.4064/sm-143-2-121-144


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.


  • A. Böttcher
  • M. Seybold

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