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.