Banach Algebras with Symbol and Singular Integral Operators

ISBN-13: 9783034854658 / Angielski / Miękka / 2014 / 206 str.

Naum Krupnik

Opis

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.