Krein–Rutman theorem
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 formula_1 be a Banach space, and let formula_2 be a convex cone such that formula_3, and formula_4 is dense in formula_1, i.e. the closure of the set formula_6. formula_7 is also known as a total cone. Let formula_8 be a non-zero compact operator which is "positive", meaning that formula_9, and assume that its spectral radius formula_10 is strictly positive. Then formula_10 is an eigenvalue of  formula_12 with positive eigenvector, meaning that there exists formula_13 such that formula_14. De Pagter's theorem. If the positive operator formula_12 is assumed to be ideal "irreducible", namely, there is no ideal formula_16,formula_1 such that formula_18, then de Pagter's theorem asserts that formula_19. Therefore, for ideal irreducible operators the assumption formula_19 is not needed.