Cent. Eur. J. Math. • 10(6) • 2012 • 2240-2263 DOI: 10.2478/s11533-012-0118-3 Perron–Frobenius and Krein–Rutman theorems for tangentially positive operators Adam Kanigowski 1∗ , Wojciech Kryszewski 1† 1 Faculty of Mathematics and Computer Sciences, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland We study several aspects of a generalized Perron–Frobenius and Krein–Rutman theorems concerning spectral properties of a (possibly unbounded) linear operator on a cone in a Banach space. The operator is subject to the so-called tangency or weak range assumptions implying the resolvent invariance of the cone. The further assumptions rely on relations between the spectral and essential spectral bounds of the operator. In general we do not assume that the cone induces the Banach lattice structure into the underlying space. 47A10, 47A75, 47D03, 15A18 Eigenvalue • Eigenvector • Spectral bound • Essential spectrum • Positive operators • Tangent cone • Tangency condition • Perron–Frobenius theorem • Krein–Rutman theorem • Strongly continuous semigroup © Versita Sp. z o.o. 1. Introduction ThecelebratedPerrontheoremassertsthatthespectralradius r(A)ofanonnegativematrix A =[a ij ] i,j =1,...,n ∈ R n×n ,i.e., a ij  0, i, j =1,...,n,isaneigenvalueof A towhichtherecorrespondsanonnegativeeigenvector x =(x 1 ,...,x n ) ∈ R n , i.e., x i  0, i =1,...,n. Observe that if K = R n + def = {x =(x 1 ,...,x n ): x i  0,i =1,...,n} is the nonnegative orthant in R n and A =[a ij ] ∈ R n×n , then a ij  0, i, j =1,...,n, if and only if A(K ) ⊂K . The well-known Krein–Rutman theorem states that given a bounded and compact linear operator A : E → E, where E is a Banach space, if A(K ) ⊂K , where K isatotalconein E, then the spectral radius r(A) belongs to the point spectrum σ p (A)of A and there is x∈K such that Ax = r(A) x provided r(A) > 0. ∗ E-mail: hannibal@mat.umk.pl † E-mail: wkrysz@mat.umk.pl Unauthenticated Download Date | 7/26/18 6:38 PM