Krein–Rutman theorem

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.

be a Banach space, and let

be a convex cone such that

, i.e. the closure of the set

is also known as a total cone.

be a non-zero compact operator, and assume that it is positive, meaning that

, and that its spectral radius

is strictly positive.

with positive eigenvector, meaning that there exists

If the positive operator

is assumed to be ideal irreducible, namely, there is no ideal

, then de Pagter's theorem[3] asserts that

Therefore, for ideal irreducible operators the assumption