The structure of the algebraic eigenspace to the spectral radius of eventually compact, nonnegative integral operators

Abstract

Let K be an eventually compact linear integral operator on L^p(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. This work extends the previous analysis of the author who characterized the distinguished eigenvalues of K and K^∗, and the support sets for the eigenfunctions and generalized eigenfunctions belonging to the spectral radius of K or K^∗. The characterizations of the support sets for the algebraic eigenspaces of K or K^∗ are phrased in terms of significant k-components which are maximal irreducible subsets of Ω and which yield a positive spectral radius for the integral operator defined by the restriction of k(x, y) to the Cartesian product of such sets. In this paper, we show that a basis for the functions, constituting the algebraic eigenspaces of K and K^∗ belonging to the spectral radius of K, can be chosen to consist of elements which are positive on their sets of support, except possibly on sets of measure less than some arbitrarily specified positive number. In addition, we present necessary and sufficient conditions, in terms of the significant k-components, for both K and K^∗ to possess a positive eigenfunction (a.e. μ) corresponding to the spectral radius, as well as necessary and sufficient conditions for the sequence ∥ γⁿKⁿg ∥_p to converge whenever g⩾ 0, where ∥ − ∥_p denotes the norm in $\text{L}{}^{\mathrm{p}}\text{(Ω, μ)}$, and γ₁ the smallest (in modulus) characteristic value of K. This analysis is made possible by introducing the concepts of chains, lengths of chains, height, and depth of a significant k-component as was done by U. Rothblum [Lin. Alg. Appl. 12 (1975), 281–292] for the matrix setting.