Banach Algebras with Symbol and Singular Integral Operators

✍ Scribed by Prof. Naum Yakovlevich Krupnik (auth.)

• Publisher: Birkhäuser Basel
• Year: 1987
• Tongue: English
• Leaves: 212
• Series: Operator Theory: Advances and Applications 26
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages I-X
Matrix Fredholm Operators....Pages 1-19
Exact Constants in Boundedness Theorems for Singular Integral Operators....Pages 21-66
Singular Integral Operators with Matrix Coefficients....Pages 67-90
Banach Algebras with Symbol....Pages 91-119
Banach Algebras Generated by Singular Integral Operators with Piecewise-Continuous Coefficients....Pages 121-143
Sufficient Families of n -Dimensional Representations of Banach Algebras....Pages 145-179
Matrix Symbols....Pages 181-198
Back Matter....Pages 199-206

✦ Subjects

Science, general