Krein–Rutman theorem
A generalization of the Perron–Frobenius theorem to Banach spaces From Wikipedia, the free encyclopedia
In functional analysis, the Krein–Rutman theorem is a generalisation of the Perron–Frobenius theorem to infinite-dimensional Banach spaces.[1] It was proved by Krein and Rutman in 1948.[2]
Statement
Let be a Banach space, and let be a convex cone such that , and is dense in , i.e. the closure of the set . is also known as a total cone. Let be a non-zero compact operator, and assume that it is positive, meaning that , and that its spectral radius is strictly positive.
Then is an eigenvalue of with positive eigenvector, meaning that there exists such that .
De Pagter's theorem
If the positive operator is assumed to be ideal irreducible, namely, there is no ideal of such that , then de Pagter's theorem[3] asserts that .
Therefore, for ideal irreducible operators the assumption is not needed.
References
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