Invariant subspaces for Fréchet spaces without continuous norm

Let ( X , p j stretchy="false">) (X,(p_j)) be a Fréchet space with Schauder basis and without continuous norm, where encoding="application/x-tex">(p_j) is an increasing sequence of seminorms inducing the topology alttext="upper X"> encoding="application/x-tex">X . We show that satisfies Invariant Subspace Property if only there exists alttext="j 0 greater-than-or-equal-to 1"> 0 ≥ 1 encoding="application/x-tex">j_0\ge 1 such alttext="kernel plus ker ⁡<!-- ⁡ <mml:mrow class="MJX-TeXAtom-ORD"> + encoding="application/x-tex">\ker p_{j+1} finite codimension in j"> p_{j} for every 0"> encoding="application/x-tex">j\ge j_0