Krein–Rutman theorem
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, 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 asserts that . Therefore, for ideal irreducible operators the assumption  is not needed. References 

Spectral theory
Theorems in functional analysis