On the ideal structure of positive, eventually compact linear operators on Banach lattices (Q811615)

We study the structure of the algebraic eigenspace corresponding to the spectral radius of a nonnegative reducible linear operator, having a completely continuous 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.