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Birkhäuser

Banach Algebras with Symbol and Singular Integral Operators

  • Book
  • © 1987

Overview

Authors:
  1. Naum Yakovlevich Krupnik

    1. Kishinev State University, Kishinev, USSR

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Part of the book series: Operator Theory: Advances and Applications (OT, volume 26)

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Table of contents (7 chapters)

  1. Front Matter

    Pages I-X
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  2. Matrix Fredholm Operators

    • Naum Yakovlevich Krupnik
    Pages 1-19

  3. Exact Constants in Boundedness Theorems for Singular Integral Operators

    • Naum Yakovlevich Krupnik
    Pages 21-66

  4. Singular Integral Operators with Matrix Coefficients

    • Naum Yakovlevich Krupnik
    Pages 67-90

  5. Banach Algebras with Symbol

    • Naum Yakovlevich Krupnik
    Pages 91-119

  6. Banach Algebras Generated by Singular Integral Operators with Piecewise-Continuous Coefficients

    • Naum Yakovlevich Krupnik
    Pages 121-143

  7. Sufficient Families of n-Dimensional Representations of Banach Algebras

    • Naum Yakovlevich Krupnik
    Pages 145-179

  8. Matrix Symbols

    • Naum Yakovlevich Krupnik
    Pages 181-198

  9. Back Matter

    Pages 199-206
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Keywords

About this book

About fifty years aga S. G. Mikhlin, in solving the regularization problem for two-dimensional singular integral operators [56], assigned to each such operator a func­ tion which he called a symbol, and showed that regularization is possible if the infimum of the modulus of the symbol is positive. Later, the notion of a symbol was extended to multidimensional singular integral operators (of arbitrary dimension) [57, 58, 21, 22]. Subsequently, the synthesis of singular integral, and differential operators [2, 8, 9]led to the theory of pseudodifferential operators [17, 35] (see also [35(1)-35(17)]*), which are naturally characterized by their symbols. An important role in the construction of symbols for many classes of operators was played by Gelfand's theory of maximal ideals of Banach algebras [201. Using this the­ ory, criteria were obtained for Fredholmness of one-dimensional singular integral operators with continuous coefficients [34 (42)], Wiener-Hopf operators [37], and multidimensional singular integral operators [38 (2)]. The investigation of systems of equations involving such operators has led to the notion of matrix symbol [59, 12 (14), 39, 41]. This notion plays an essential role not only for systems, but also for singular integral operators with piecewise-continuous (scalar) coefficients [44 (4)]. At the same time, attempts to introduce a (scalar or matrix) symbol for other algebras have failed.

Authors and Affiliations

  • Kishinev State University, Kishinev, USSR

    Naum Yakovlevich Krupnik

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