In functional analysis, the 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 \( {\displaystyle K-K} \) is dense in X, i.e. the closure of the set \( {\displaystyle \{u-v:u,\,v\in K\}=X} \). K is also known as a total cone. Let \( {\displaystyle T:X\to X} \) be a non-zero compact operator which is positive, meaning that \( {\displaystyle T(K)\subset K} \), and assume that its spectral radius \( {\displaystyle r(T)} \) is strictly positive.
Then \( {\displaystyle r(T)} \) is an eigenvalue of 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 T is assumed to be ideal irreducible, namely, there is no ideal \( {\displaystyle J\neq 0} \), 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
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.
Functional analysis (topics – glossary)
Spaces
Hilbert space Banach space Fréchet space topological vector space
Theorems
Hahn–Banach theorem closed graph theorem uniform boundedness principle Kakutani fixed-point theorem Krein–Milman theorem min-max theorem Gelfand–Naimark theorem Banach–Alaoglu theorem
Operators
bounded operator compact operator adjoint operator unitary operator Hilbert–Schmidt operator trace class unbounded operator
Algebras
Banach algebra C*-algebra spectrum of a C*-algebra operator algebra group algebra of a locally compact group von Neumann algebra
Open problems
invariant subspace problem Mahler's conjecture
Applications
Besov space Hardy space spectral theory of ordinary differential equations heat kernel index theorem calculus of variation functional calculus integral operator Jones polynomial topological quantum field theory noncommutative geometry Riemann hypothesis
Advanced topics
locally convex space approximation property balanced set Schwartz space weak topology barrelled space Banach–Mazur distance Tomita–Takesaki theory
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 of a spectrum
(Continuous Point Residual) Approximate point Compression Discrete Spectral abscissa
Spectral Theorem
Borel functional calculus Min-max theorem Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras
Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Uniform algebra
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 Limiting absorption principle Unbounded operator
Examples
Wiener algebra
Applications
Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law
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 \( {\displaystyle K-K} \) is dense in X, i.e. the closure of the set \( {\displaystyle \{u-v:u,\,v\in K\}=X} \). K is also known as a total cone. Let \( {\displaystyle T:X\to X} \) be a non-zero compact operator which is positive, meaning that \( {\displaystyle T(K)\subset K} \), and assume that its spectral radius \( {\displaystyle r(T)} \) is strictly positive.
Then \( {\displaystyle r(T)} \) is an eigenvalue of 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 T is assumed to be ideal irreducible, namely, there is no ideal \( {\displaystyle J\neq 0} \), 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
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.
Functional analysis (topics – glossary)
Spaces
Hilbert space Banach space Fréchet space topological vector space
Theorems
Hahn–Banach theorem closed graph theorem uniform boundedness principle Kakutani fixed-point theorem Krein–Milman theorem min-max theorem Gelfand–Naimark theorem Banach–Alaoglu theorem
Operators
bounded operator compact operator adjoint operator unitary operator Hilbert–Schmidt operator trace class unbounded operator
Algebras
Banach algebra C*-algebra spectrum of a C*-algebra operator algebra group algebra of a locally compact group von Neumann algebra
Open problems
invariant subspace problem Mahler's conjecture
Applications
Besov space Hardy space spectral theory of ordinary differential equations heat kernel index theorem calculus of variation functional calculus integral operator Jones polynomial topological quantum field theory noncommutative geometry Riemann hypothesis
Advanced topics
locally convex space approximation property balanced set Schwartz space weak topology barrelled space Banach–Mazur distance Tomita–Takesaki theory
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 of a spectrum
(Continuous Point Residual) Approximate point Compression Discrete Spectral abscissa
Spectral Theorem
Borel functional calculus Min-max theorem Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras
Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Uniform algebra
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 Limiting absorption principle Unbounded operator
Examples
Wiener algebra
Applications
Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law
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