On the Collatz-Wielandt numbers and the local spectral radius of a nonnegative matrix

Abstract

Let T be a nonnegative linear continuous operator in a partially ordered Banach space E, and let x be a nonnegative nonzero vector in E. We establish relations between the local spectral radius rT(x) and the upper and lower Collatz-Wielandt numbers rT(x) and rT(x). Examples will show that the situation in an infinite dimensional space ∗FE can be much more complicated than in the classical case of Rn with positive cone Rn+. We also present in certain situations sufficient and/or necessary conditions for both sequences {rT(Tnx)}n and {rT(Tnx)}n to converge to rT(x). The significance of some conditions in the infinite dimensional case is demonstrated, and we make several comments on the earlier literature of the subject.