Perron-Frobenius and Krein-Rutman theorems for tangentially positive operators

• Published in: Central European journal of mathematics Vol. 10; no. 6; pp. 2240 - 2263
• Main Authors: Kanigowski, Adam, Kryszewski, Wojciech
• Format: Journal Article
• Language: English
• Published: Heidelberg SP Versita 01.12.2012; Versita; De Gruyter Brill Sp. z o.o., Paradigm Publishing Services; De Gruyter
• Subjects: 15A18; 47A10; 47A75; 47D03; Algebra; Banach spaces; Eigenvalue; Eigenvector; Essential spectrum; Geometry; Krein-Rutman theorem; Lie Groups; Linear operators; Mathematics; Mathematics and Statistics; Number Theory; Perron-Frobenius theorem; Positive operators; Probability Theory and Stochastic Processes; Research Article; Spectra; Spectral bound; Strongly continuous semigroup; Tangency condition; Tangent cone; Theorems; Topological Groups; Tangency condition; Eigenvector; Perron-Frobenius theorem; Eigenvalue; Spectral bound; 47A75; 47A10; Positive operators; Tangent cone; 15A18; Krein-Rutman theorem; 47D03; Essential spectrum; Strongly continuous semigroup
• ISSN: 1895-1074; 2391-5455; 1644-3616; 2391-5455
• DOI: 10.2478/s11533-012-0118-3

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.