Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-28T02:50:02.167Z Has data issue: false hasContentIssue false
  • English
  • Français

An inverse for the Gohberg-Krupnik symbol map

Published online by Cambridge University Press:  14 November 2011

Martin Costabel
Martin Costabel
Affiliation:
Fachbereich Mathematik der Technischen Hochschule Darmstadt, Germany
Get access Rights & Permissions [Opens in a new window]

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
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 87 , Issue 1-2 , 1980 , pp. 153 - 165
Copyright
Copyright © Royal Society of Edinburgh 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Boutet de Monvel, L.. Boundary problems for pseudo-differential operators. Acta Math. 126 (1971), 11–51.CrossRefGoogle Scholar
2Cordes, H. O.. Pseudo-differential operators on a half-line. J. Math. Mech. 18 (1969), 893–908.Google Scholar
3Costabel, M.. A contribution to the theory of singular integral equations with Carleman shift. Integral Equations Oper. Theory 2 (1979), 12–24.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), 515–537.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), 475–486.CrossRefGoogle Scholar
8Gerlach, E. and Kremer, M.. Singulare Integraloperatoren in Lp-Räumen. Math. Ann. 204 (1973), 285–304.CrossRefGoogle Scholar
9Gohberg, I. C. and Ja, N.. Krupnik. Singular integral operators with piecewise continuous coefficients and their symbols. Math. USSR-Izv. 5 (1971), 955–979.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), 279–293.CrossRefGoogle Scholar
13Speck, F. O.. Über verallgemeinerte Faltungsoperatoren und eine Klasse von Integrodifferential-gleichungen (Darmstadt: Dissertation, 1974).Google Scholar