# Banach Algebras with Symbol and Singular Integral Operators

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

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

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.

algebra character construction differential operator equation Matrix Maxima maximum Natural operator pseudodifferential operator pseudodifferential operators Scala Singular integral time

- DOI https://doi.org/10.1007/978-3-0348-5463-4
- Copyright Information Birkhäuser Basel 1987
- Publisher Name Birkhäuser, Basel
- eBook Packages Springer Book Archive
- Print ISBN 978-3-0348-5465-8
- Online ISBN 978-3-0348-5463-4
- Series Print ISSN 0255-0156
- Series Online ISSN 2296-4878
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