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.^[2]

Statement

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

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

De Pagter's theorem

If the positive operator ${\displaystyle T}$ is assumed to be ideal irreducible, namely, there is no ideal ${\displaystyle J\neq 0}$ of ${\displaystyle X}$ such that ${\displaystyle TJ\subset J}$, then de Pagter's theorem^[3] asserts that ${\displaystyle r(T)>0}$.

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

References