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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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3 Answers

up vote 6 down vote accepted

"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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Thank you! This is exactly what I needed. –  Mark Kim Jun 16 '11 at 0:22
You're welcome. –  Andrey Rekalo Jun 16 '11 at 0:24
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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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