In functional analysis, the Krein–Rutman theorem is a generalisation of the Perron–Frobenius theorem to infinite-dimensional Banach spaces. It was proved by Krein and Rutman in 1948.
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Statement
Let X be a Banach space, and let K⊂X be a convex cone such that K-K is dense in X. Let T:X→X be a non-zero compact operator which is positive, meaning that T(K)⊂K, and assume that its spectral radius r(T) is strictly positive.
Then r(T) is an eigenvalue of T with positive eigenvector, meaning that there exists u∈K0 such that T(u)=r(T)u.
De Pagter's theorem
If the positive operator T is assumed to be ideal irreducible, namely, there is no ideal J≠0,X such that T J⊂J, then de Pagter's theorem asserts that r(T)>0.
Therefore, for ideal irreducible operators the assumption r(T)>0 is not needed.