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25995 results in Abstract analysis

Page 175 of 1300

  • 5 - Toeplitz Matrices: Moments, Spectra, Asymptotics

  • Nikolaï Nikolski, Université de Bordeaux
  • Translated by Danièle Gibbons, Greg Gibbons
  • Book:
    Toeplitz Matrices and Operators
    Published online:
    06 December 2019
    Print publication:
    02 January 2020, pp 230-328

  • Summary

    Trigonometric moment problem, positive definite Toeplitz matrices, orthogonal polynomials.

    Arveson’s distance formula, the four-block lemma, Julia’s lemma.

    Inversion of a Toeplitz matrix by Gohberg-Sementsul and Baxter-Hirschman, inversion by the finite section method.

    Circulants.

    Toeplitz determinants and their asymptotics, equidistribution following Weyl, asymptotic distribution of Toeplitz spectra (following Szegő) and of circulants.

    The “strong Szegő theorem,” Krein-Dirichlet algebra.

    The Helton-Howe and Berger-Shaw formulas of det/trace.

    Positive harmonic functions, holomorphic functions of the Schur class

    Inversion of Wiener–Hopf operators by the finite section method, case of a rational symbol.

    Cauchy and Hilbert determinants, Libkind-Widom and Borodin Okounkov formulas.

  • 1 - Why Toeplitz–Hankel? Motivations and Panorama

  • Nikolaï Nikolski, Université de Bordeaux
  • Translated by Danièle Gibbons, Greg Gibbons
  • Book:
    Toeplitz Matrices and Operators
    Published online:
    06 December 2019
    Print publication:
    02 January 2020, pp 1-30

  • Summary

    Four cornerstones of the theory of Toeplitz operators: the Riemann–Hilbert problem (RHP), the singular integral operators (SIO), the Wiener–Hopf operators (WHO), and last (but not least) the Toeplitz matrices and operators (TMO) (strictly speaking, compressions of multiplication operators).

    The founding contributions of Bernhard Riemann, David Hilbert, George Birkhoff, Otto Toeplitz, Gábor Szegő, Norbert Wiener, and Eberhard Hopf.

    The modern and post-modern periods of the theory.

  • 3 - H2 Theory of Toeplitz Operators

  • Nikolaï Nikolski, Université de Bordeaux
  • Translated by Danièle Gibbons, Greg Gibbons
  • Book:
    Toeplitz Matrices and Operators
    Published online:
    06 December 2019
    Print publication:
    02 January 2020, pp 101-177

  • Summary

    The Toeplitz algebra, semi-commutators and symbolic calculus, index and Cauchy topological index, homotopy of symbols.

    dist(f, H∞) and dist(f, H∞ + C), local principle, local distances.

    The spectrum and essential spectrum of Toeplitz operators, sectorial and locally sectorial symbols, spectral inclusions.

    Wiener–Hopf factorization.

    Self-adjoint Toeplitz operators, Hankel operators in the Toeplitz algebra.

Page 175 of 1300