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The classical Krein-Rutman theorem states that any positive compact linear endomorphism $T:X \to X$ on a Banach space $X$ with positive spectral radius $r(T)$ has an eigenvalue $r(T)$ with a positive eigenvector. Papers and textbooks seem to write off the theorem as "standard" and "well-known", but I have not been able to locate any exposition with a proof of the theorem.

Is there a reference (preferably a textbook) including the statement and the proof of the theorem? (The original paper of Krein and Rutman appears to be in Russian, which I cannot read.)

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"Topological Vector Spaces" by Helmut Schaefer contains a thorough treatment of the classical Krein-Rutman theorem for compact positive operators in an ordered Banach space along with several generalizations to the case of a locally convex space with a cone. See Section 2 of the Appendix, Pringsheim's Theorem and Its Consequences. .

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For Banach lattices, a statement and a sketch of the proof can be found in Abramovich and Aliprantis's article "Positive operators" in the Handbook of the Geometry of Banach Spaces: see Google books excerpt

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Here is a paper with a proof that is available online: http://www.mat.umk.pl/c/document_library/get_file?uuid=9019c4ed-5803-4427-b32a-4c64f6b1d83a&groupId=671

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  • $\begingroup$ The link is broke. Could you please write out the full reference which is independent of a link and post a new working link? Thank you. $\endgroup$
    – Hans
    Commented Jul 12, 2018 at 8:05
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    $\begingroup$ No. I do not remember that seven year later! $\endgroup$
    – Michael Renardy
    Commented Jul 12, 2018 at 17:47

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