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An inverse for the Gohberg-Krupnik symbol map

Published online by Cambridge University Press:  14 November 2011

Martin Costabel
Affiliation:
Fachbereich Mathematik der Technischen Hochschule Darmstadt, Germany

Synopsis

It is shown that the elements of the closed operator algebra generated by one-dimensional singular integral operators with piecewise continuous coefficients with a fixed finite set of points of discontinuity can be written as the sum of a singular integral operator, a compact operator, and generalized Mellin convolutions. Their Gohberg-Krupnik symbol is given in terms of the Mellin transform. This gives an explicit construction of an operator with prescribed Gohberg—Krupnik symbol.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1980

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References

1Boutet de Monvel, L.. Boundary problems for pseudo-differential operators. Acta Math. 126 (1971), 1151.CrossRefGoogle Scholar
2Cordes, H. O.. Pseudo-differential operators on a half-line. J. Math. Mech. 18 (1969), 893908.Google Scholar
3Costabel, M.. A contribution to the theory of singular integral equations with Carleman shift. Integral Equations Oper. Theory 2 (1979), 1224.CrossRefGoogle Scholar
4Costabel, M.. Singulare Integraloperatoren auf Kurven mit Ecken (Darmstadt: THD Preprint 483, 1979).Google Scholar
5Dudučava, R. V.. On bisingular integral operators with discontinuous coefficients. Math. USSR-Sb. 30 (1976), 515537.CrossRefGoogle Scholar
6Eskin, G. I.. Boundary value problems for elliptic pseudo differential equations (russian) (Moscow: Nauka, 1973).Google Scholar
7Figà-Talamanca, A. and Gaudry, G. I.. Multipliers of Lp which vanish at infinity. J. Functional Analysis 7 (1971), 475486.CrossRefGoogle Scholar
8Gerlach, E. and Kremer, M.. Singulare Integraloperatoren in Lp-Räumen. Math. Ann. 204 (1973), 285304.CrossRefGoogle Scholar
9Gohberg, I. C. and Ja, N.. Krupnik. Singular integral operators with piecewise continuous coefficients and their symbols. Math. USSR-Izv. 5 (1971), 955979.CrossRefGoogle Scholar
10Gohberg, I. C. and Ja, N.. Krupnik. Einführung in die Theorie der eindimensionalen singulären Integraloperatoren (Basel: Birkhäuser, 1979).CrossRefGoogle Scholar
11Jörgens, K.. Lineare Integraloperatoren (Stuttgart: Teubner, 1970).CrossRefGoogle Scholar
12Simonenko, I. B.. Operators of convolution type in cones. Math. USSR-Sb. 3 (1967), 279293.CrossRefGoogle Scholar
13Speck, F. O.. Über verallgemeinerte Faltungsoperatoren und eine Klasse von Integrodifferential-gleichungen (Darmstadt: Dissertation, 1974).Google Scholar