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.
