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 $$X$$ be a Banach space, and let $$K\subset X$$ be a convex cone such that $$K\cap -K = \{0\}$$, and $$K-K$$ is dense in $$X$$, i.e. the closure of the set $$\{u - v : u,\,v\in K\}=X$$. $$K$$ is also known as a total cone. Let $$T:X\to X$$ be a non-zero compact operator, and assume that it is positive, meaning that $$T(K)\subset K$$, and that its spectral radius $$r(T)$$ is strictly positive.

Then $$r(T)$$ is an eigenvalue of $$T$$ with positive eigenvector, meaning that there exists $$u\in K\setminus {0}$$ such that $$T(u)=r(T)u$$.

De Pagter's theorem
If the positive operator $$T$$ is assumed to be ideal irreducible, namely, there is no ideal $$J\ne0$$ of $$X$$ such that $$TJ \subset J$$, then de Pagter's theorem asserts that $$r(T)>0$$.

Therefore, for ideal irreducible operators the assumption $$r(T)>0$$ is not needed.