Abschätzungen für die eigenwerte positiver linearer operatoren

ZusammenfassungThe problem treated is to upper bounds for |λ(A)|π(A), where A is a positive matrix, λ(A) is any one of its eigenvalues and π(A) is the biggest eigenvalue. Although Hopf's inequality [7] is in a certain sense best possible, the inequality can be improved if one makes use of the eigenvector associated with π(A). The general setting is that of a linear operator A (in a locally convex space) mapping a cone into itself; the technique is to introduce semi-norms in such a way that the contraction of the cone measured in these norms gives an upper bound for |λ(A)|π(A). In Sect. 5 the results are applied to non-nagative matrices. A converse problem is treated in Sect. 6