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

Authors: ELİF UYANIK, MURAT HAYRETTİN YURDAKUL

Abstract: Let $\ell$ 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 $\ell$-Köthe spaces, then there exists a continuous unbounded quasidiagonal operator between them. Using this result, we study the corresponding Köthe matrices when every continuous linear operator between $\ell$-Köthe spaces is bounded. As an application, we observe that the existence of an unbounded operator between $\ell$-Köthe spaces, under a splitting condition, causes the existence of a common basic subspace.

Keywords: Bounded operators, unbounded operators, $\ell$-Köthe spaces

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