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Product-integration and one-parameter subgroups of linear lie-groups

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1163)

Abstract

The pilgerschritt transform is an iterative method which can be used to calculate one-parameter subgroups of groups of matrices, whose restrictions to the interval [0,1] belongs to a given homotopy class. The restrictions of one-parameter subgroups are fixed points of this method. All fixed points can be described by a product integral equation. Until now there could be shown for special groups of matrices that these fixed points are attractive. This paper deals with the group Aff(1,ℝ), where the product integral equation can be reduced to a Fredholm integral equation of the second kind. In this case the existence of one-parameter subgroups is proved, whose restrictions are not attractive fixed points. Furthermore, the existence of other fixed points is shown.

Presented by N. Netzer

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References

  1. Dollard, J.D., and Friedman, C.N.: "Product Integration". Encyclopedia of mathematics and its applications, Addison-Wesley, London, 1979.

    Google Scholar 

  2. Dollard, J.D., and Friedman, C.N.: On strong product integration. J. Func. Anal. 28, 309–354 (1978).

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Dollard, J.D., and Friedman, C.N.: Product integrals II: Contour integrals. J. Func. Anal. 28, 355–368 (1978).

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Förg-Rob, W., and Netzer, N.: Eine Methode zur Berechnung von einparametrigen Untergruppen ohne Verwendung des Logarithmus. Sitzungsberichte der Österreichischen Akademie der Wissenschaften II/190, 273–284 (1981).

    MathSciNet  MATH  Google Scholar 

  5. Gohberg, I.C., and Krein, M.G.: "Introduction to the Theory of Linear Nonselfadjoint Operators". Translations of Mathematical Monographs, vol. 18, American Mathematical Society, Providence, R.I., 1969.

    Google Scholar 

  6. Guggenheimer, H.W.: "Differential Geometry". Mac Graw-Hill, New York, 1963.

    MATH  Google Scholar 

  7. Hoheisl, G.: "Integralgleichungen". Sammlung GÖschen Band Nr. 1099, Berlin-Leipzig, 1936.

    Google Scholar 

  8. Hostinsky, B., and Volterra, V.: "Operations Infinitesimales Lineaires", Paris, 1938.

    Google Scholar 

  9. Kuhnert, K.: Die Konvergenz des Pilgerschrittverfahrens für unipotente und auflösbare lineare Gruppen. Berichte der mathematisch-statistischen Sektion im Forschungszentrum Graz Nr. 87 (1978).

    Google Scholar 

  10. Liedl, R.: Über eine Methode zur Lösung der Translationsgleichung. Berichte der mathematisch-statistischen Sektion im Forschungszentrum Graz, Nr.84 (1978).

    Google Scholar 

  11. Liedl, R., Netzer, N., Reitberger, H.: Über eine Methode zur Auffindung stetiger Iterationen in Lie-Gruppen, Aequationes Mathematicae 24, 19–32 (1982).

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Netzer, N., and Reitberger, H.: On the convergence of iterated pilgerschritt transformations in nilpotent Lie Groups. Publicationes Mathematicae Debrecen 29, 309–314 (1982).

    MathSciNet  MATH  Google Scholar 

  13. Schlesinger, L.: Neue Grundlagen für einen Infinitesimalkalkül der Matrizen. Math. Zeit. 33, 33–61 (1931).

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Zabreyko, P.P. et al.: "Integral equations-a reference text". Noordhoff International Publishing Leyden, 1975.

    Google Scholar 

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© 1985 Springer-Verlag

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Förg-Rob, W., Netzer, N. (1985). Product-integration and one-parameter subgroups of linear lie-groups. In: Liedl, R., Reich, L., Targonski, G. (eds) Iteration Theory and its Functional Equations. Lecture Notes in Mathematics, vol 1163. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076419

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  • DOI: https://doi.org/10.1007/BFb0076419

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16067-0

  • Online ISBN: 978-3-540-39749-6

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