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 an unconditional basis. We show that if there exists a continuous linear unbounded operator between l-Köthe spaces, then there exists a continuous unbounded quasi-diagonal operator between them. Using this result, we study in terms of corresponding Köthe matrices when every continuous linear operator between l-Köthe spaces is bounded. As an application, we observe that the existence of an unbounded operator between l-Köthe spaces, under a splitting condition, causes the existence of a common basic subspace.

arXiv comment6 pages