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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
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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.


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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

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