INVARIANT SUBSPACES FOR FRECHET SPACES WITHOUT CONTINUOUS NORM

Let (X, (p(j))) be a Frechet space with a Schauder basis and without continuous norm, where (p(j)) is an increasing sequence of seminorms inducing the topology of X. We show that X satisfies the Invariant Subspace Property if and only if there exists j(0) >= 1 such that ker p(j+1) is of finite codimension in ker p(j) for every j >= j(0).