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. 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 which is positive, meaning that , and assume 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 asserts that . Therefore, for ideal irreducible operators the assumption is not needed. References Spectral theory Theorems in functional analysis