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 $X$ be a Banach space, and let $K\subset X$ be a convex cone such that $K-K$ 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 which is positive, meaning that $T(K)\subset K$ , and assume 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 $J\neq 0$ , $X$ 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

^ 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". Uspehi Matem. Nauk (N. S.) (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". Uspehi Matem. Nauk (N. S.) (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
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
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
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
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Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
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