A remark on a paper of P. B. Djakov and M. S. Ramanujan

Abstract

Let ℓ be a Banach sequence space with a monotone norm in which the canonical system $(e_n)$ is anunconditional basis. We show that if there exists a continuous linear unbounded operator between ℓ -Köthe spaces, thenthere exists a continuous unbounded quasidiagonal operator between them. Using this result, we study the correspondingKöthe matrices when every continuous linear operator between ℓ -Köthe spaces is bounded. As an application, we observethat the existence of an unbounded operator between ℓ -Köthe spaces, under a splitting condition, causes the existenceof a common basic subspace.