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[edit]

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[edit]

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

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

References[edit]

^ Du, Y. (2006). "1. Krein–Rutman Theorem and the Principal Eigenvalue". Order structure and topological methods in nonlinear partial differential equations. Vol. 1. Maximum principles and applications. Series in Partial Differential Equations and Applications. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd. ISBN 981-256-624-4. MR 2205529.
^ Kreĭn, M.G.; Rutman, M.A. (1948). "Linear operators leaving invariant a cone in a Banach space". Uspekhi Mat. Nauk. New Series (in Russian). 3 (1(23)): 1–95. MR 0027128.. English translation: Kreĭn, M.G.; Rutman, M.A. (1950). "Linear operators leaving invariant a cone in a Banach space". Amer. Math. Soc. Transl. 1950 (26). MR 0038008.
^ de Pagter, B. (1986). "Irreducible compact operators". Math. Z. 192 (1): 149–153. doi:10.1007/bf01162028. MR 0835399.

[1] Du, Y. (2006). "1. Krein–Rutman Theorem and the Principal Eigenvalue". Order structure and topological methods in nonlinear partial differential equations. Vol. 1. Maximum principles and applications. Series in Partial Differential Equations and Applications. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd. ISBN 981-256-624-4. MR 2205529.

[2] Kreĭn, M.G.; Rutman, M.A. (1948). "Linear operators leaving invariant a cone in a Banach space". Uspekhi Mat. Nauk. New Series (in Russian). 3 (1(23)): 1–95. MR 0027128.. English translation: Kreĭn, M.G.; Rutman, M.A. (1950). "Linear operators leaving invariant a cone in a Banach space". Amer. Math. Soc. Transl. 1950 (26). MR 0038008.

[3] Pagter, B. (1986). "Irreducible compact operators". Math. Z. 192 (1): 149–153. doi:10.1007/bf01162028. MR 0835399.

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v t e Spectral theory and ^*-algebras
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Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
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