Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Positive solutions of positive linear equations


Author: Paul Nelson
Journal: Proc. Amer. Math. Soc. 31 (1972), 453-457
DOI: https://doi.org/10.1090/S0002-9939-1972-0288605-5
MathSciNet review: 0288605
Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: Let $ B$ be a real vector lattice and a Banach space under a semimonotonic norm. Suppose $ T$ is a linear operator on $ B$ which is positive and eventually compact, $ y$ is a positive vector, and $ \lambda $ is a positive real. It is shown that $ {(\lambda I - T)^{ - 1}}y$ is positive if, and only if, $ y$ is annihilated by the absolute value of any generalized eigenvector of $ {T^\ast}$ associated with a strictly positive eigenvalue not less than $ \lambda $. A strictly positive eigenvalue is a positive eigenvalue having an associated positive eigenvector. For the case of $ B = {L^p}$ this yields the result that $ {(\lambda I - T)^{ - 1}}y \geqq 0$ if, and only if, $ y$ is almost everywhere zero on a certain set which depends on $ \lambda $ but is otherwise fixed.


References [Enhancements On Off] (What's this?)

  • [1] Mahlon M. Day, Normed linear spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. Heft 21. Reihe: Reelle Funktionen, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0094675
  • [2] M. A. Krasnosel′skiĭ, Positive solutions of operator equations, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron, P. Noordhoff Ltd. Groningen, 1964. MR 0181881
  • [3] Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950. MR 0033869
  • [4] Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0098966
  • [5] M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspehi Matem. Nauk (N. S.) 3 (1948), no. 1(23), 3–95 (Russian). MR 0027128
  • [6] P. Nelson, Jr., An investigation of criticality for energy-dependent transport in slab geometry, Ph.D. Dissertation, University of New Mexico, Albuquerque, New Mexico, 1969.
  • [7] Paul Nelson Jr., Subcriticality for transport of multiplying particles in a slab, J. Math. Anal. Appl. 35 (1971), 90–104. MR 0300588, https://doi.org/10.1016/0022-247X(71)90238-1
  • [8] Samuel Karlin, Positive operators, J. Math. Mech. 8 (1959), 907–937. MR 0114138
  • [9] Helmut H. Schaefer, Topological vector spaces, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1966. MR 0193469
  • [10] A. C. Zaanen, Linear analysis, North-Holland, Amsterdam, 1964.


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0288605-5
Keywords: Ordered Banach space, positive operator, positive solution, linear operator, neutron transport, radiative transfer, positive kernel
Article copyright: © Copyright 1972 American Mathematical Society