Krein–Rutman theorem

Short description: A generalization of the Perron–Frobenius theorem to Banach spaces

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

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

De Pagter's theorem

If the positive operator [math]\displaystyle{ T }[/math] is assumed to be ideal irreducible, namely, there is no ideal [math]\displaystyle{ J\ne0 }[/math] of [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ TJ \subset J }[/math], then de Pagter's theorem asserts that [math]\displaystyle{ r(T)\gt 0 }[/math].

Therefore, for ideal irreducible operators the assumption [math]\displaystyle{ r(T)\gt 0 }[/math] is not needed.