Numerische Mathematik

, Volume 34, Issue 4, pp 403–409 | Cite as

A generalization of the Stein-Rosenberg theorem to Banach spaces

  • J. P. Milaszewicz


In the first part of this note we prove a generalization of the Stein-Rosenberg theorem; the context is that of real Banach spaces with a normal reproducing cone and the operators involved are positive and completely continuous. Our generalization of the Stein-Rosenberg theorem improves the modern version of it as stated by F. Robert in [5, §2]. In the second part, we discuss briefly how our results are related to other versions of the Stein-Rosenberg theorem. In the last section we describe a situation to which the results in the first part can be applied.

Subject Classifications

AMS(MOS):65F 10, 47B 55 CR: 5.14 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Birkhoff, G., Varga, R.S.: Reactor criticality and nonnegative matrices. SIAM J. Appl. Math.6, 354–377 (1958)Google Scholar
  2. 2.
    Fiedler, M., Pták, V.: Über die Konvergenz des verallgemeinerten Seidelschen Verfahrens zur Lösung von Systemen linearer Gleichungen. Math. Nachr.15, 31–38 (1956)Google Scholar
  3. 3.
    Krein, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a Banach space. Translations of the A.M.S., Series 1, Vol. 10, 199–325 (1962)Google Scholar
  4. 4.
    Marek, I.:u 0-positive operators and some of their applications. SIAM J. Math.15, 484–494 (1967)Google Scholar
  5. 5.
    Robert, F.: Autour du théorème de Stein-Rosenberg. Numer. Math.27, 133–141 (1976)Google Scholar
  6. 6.
    Stein, P., Rosenberg, R.L.: On the solution of linear simultaneous equations by iteration. J. London Math. Soc.23, 111–118 (1948)Google Scholar
  7. 7.
    Vandergraft, J.S.: Spectral properties of matrices which have invariant cones. SIAM J. Appl. Math.16, 1208–1222 (1968)Google Scholar
  8. 8.
    Varga, R.S.: Matrix iterative analysis, Englewood Cliffs, N.J. Prentice-Hall 1962Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • J. P. Milaszewicz
    • 1
  1. 1.Departamento de Matemática, Facultad de Ciencias Exactas y NaturalesCiudad UniversitariaBuenos AiresArgentina

Personalised recommendations