Geometric eigenspace of the generator of C₀-semigroup of positive operators

Let E be a Banach lattice with order continuous norm and {T(t)}_t≥0 be an eventually compact c₀-semigroup of positive operators on E with generator A. We investigate the structure of the geometric eigenspace of the generator belonging to the spectral bound when the semigroup is ideal reducible. It is shown that a basis of the eigenspace can be chosen to consist of elements of E with certain positivity structure. This is achieved by a decomposition of the underlying Banach lattice E into a direct sum of closed ideals which can be viewed as a generalization of the Frobenius normal form for nonnegative reducible matrices.