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Eventual Cone Invariance

By Michael Kasigwa


Thesis (Ph.D.), Mathematics, Washington State UniversityIf $K$ is a proper cone in $\RR^{n}$ some results in the theory of eventually (entrywise) nonnegative matrices have equivalent analogues in eventual $K$-invariance. We develop these analogues using the classical Perron-Frobenius theory for cone preserving maps. Using an ice-cream cone we demonstrate that unlike the entrywise nonnegative case of a matrix $A \in \m $, eventual cone invariance and eventual exponential cone invariance are not equivalent. The notions of inverse positivity of {\em M-matrices}, {\em M-type} and {\em $M_{v}$-matrices} are extended to {\em $M_{v,K}$-matrices (operators)}, that have the form $A = sI - B$, where $B$ is eventually $K$-nonnegative, that is, $B^{m}K \subseteq K$ for all sufficiently large $m$. Eventual positivity of semigroups of linear operators on Banach spaces ordered by a cone can be investigated using resolvents or resolvent type operators constructed from the generators.Washington State University, Mathematic

Topics: Mathematics, Cone-Invariance, Matrices, Nonnegative, Operator-Semigroups, Perron-Frobenius, Positive
Year: 2017
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Provided by: Research Exchange
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