On linear integral operators with nonnegative kernels

Abstract

Let K be an eventually compact linear integral operator on $\text{L}{}^{\mathrm{p}}\text{(Ω, μ), 1 ⩽ p < ∞}$, with nonnegative kernel k(x, y), where the underlying measure μ is totally σ-finite on the domain set Ω when p = 1. In considering the equation λf = Kf + g for given nonnegative $\text{g ϵ L}{}^{\mathrm{p}}\text{(Ω, μ), λ > 0}$, P. Nelson, Jr. provided necessary and sufficient conditions, in terms of the support of g, such that a nonnegative solution $\text{f ϵ L}{}^{\mathrm{p}}\text{(Ω, μ)}$ was attained. Such conditions led to generalizing some of the graph-theoretic ideas associated with the normal form of a nonnegative reducible matrix. The purpose of this paper is to show that the analysis by Nelson can be enlarged to provide a more complete generalization of the normal form of a nonnegative matrix which can be used to characterize the distinguished eigenvalues of K and K^∗, and to describe sets of support for the eigenfunctions and generalized eigenfunctions of both K and K^∗ belonging to the spectral radius of K.