On the ideal structure of positive, eventually compact linear operators on Banach lattices
Ruey-Jen Jang, H. D. Victory · Pacific Journal of Mathematics · 1993
We study the structure of the algebraic eigenspace corresponding to the spectral radius of a nonnegative reducible linear operator T, having a compact iterate and defined on a Banach lattice E with order continuous norm.The Perron-Frobenius theory is generalized by showing that this algebraic eigenspace is spanned by a basis of eigenelements and generalized eigenelements possessing certain positivity features.A combinatorial characterization of both the Riesz index of the spectral radius and the dimension of the algebraic eigenspace is given.These results are made possible by a decomposition of T, in terms of certain closed ideals of E, in a form which directly generalizes the Frobenius normal form of a nonnegative reducible matrix/