The invariant subspace problem for non-Archimedean Köthe spaces

In this study, a proof is given that if a non-Archimedean Köthe space Λ, which is generated by an infinite matrix B=(bnk)k,n∈N such that bnk≤bn+1k for k,n∈N and for each k,l∈N, m∈N exists such that bnk+1≥bn+lk for n≥m, then a continuous operator T:Λ→Λ exists that has no nontrivial closed invariant subspaces.