On the Collatz-Wielandt sets associated with a cone-preserving map
Let K be a proper cone in \(R^ n\) and \(\pi\) (K) be the set of all \(n\times n\) real matrices A such that AK\(\subseteq K\). \textit{G. P. Barker} and \textit{H. Schneider} [ibid. 11, 219-233 (1975; Zbl 0311.15011)] have introduced four sets \(\Omega,\Omega_ 1,\Sigma\) and \(\Sigma_ 1\) associated with A in \(\pi\) (K). The authors refer to these sets as Collatz-Wielandt sets. They determine the values sup \(\Omega\), sup \(\Omega_ 1\), Inf \(\Sigma\), and Inf \(\Sigma_ 1\). The given proofs are elementary and geometric and make use of the Perron-Frobenius theorem. Some known results for a nonnegative matrix relating to spectral radius to its upper and lower Collatz-Wielandt numbers are extended to the setting of a cone-preserving map. Applications of the results obtained to the nonnegativity of solutions of linear inequalities with a nonnegative matrix are also considered.
scientific article; zbMATH DE number 4120325
scientific article