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.{{cite book|mr=2205529|last=Du|first=Y.|title=Order structure and topological methods in nonlinear partial differential equations. Vol. 1. Maximum principles and applications|series=Series in Partial Differential Equations and Applications|publisher=World Scientific Publishing Co. Pte. Ltd.|location=Hackensack, NJ|year=2006|isbn=981-256-624-4|chapter=1. Krein–Rutman Theorem and the Principal Eigenvalue}} It was proved by Krein and Rutman in 1948.{{cite journal|mr=0027128|last1=Kreĭn|first1=M.G.|last2=Rutman|first2=M.A.|title=Linear operators leaving invariant a cone in a Banach space|language=Russian|journal=Uspekhi Mat. Nauk |series=New Series|volume=3|year=1948|issue=1(23)|pages=1–95}}. English translation: {{cite journal|mr=0038008|last1=Kreĭn|first1=M.G.|last2=Rutman|first2=M.A.|title=Linear operators leaving invariant a cone in a Banach space|journal=Amer. Math. Soc. Transl.|year=1950|volume=1950|issue=26}}

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{{cite journal|mr=0835399|last=de Pagter|first=B.|title=Irreducible compact operators|journal=Math. Z.|volume=192|year=1986|issue=1|pages=149–153|doi=10.1007/bf01162028}} asserts that $r(T)>0$ .

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

References

Category:Spectral theory

Category:Theorems in functional analysis