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

## TL;DR

This paper investigates the conditions under which unbounded operators exist between l-K"othe spaces, showing that such operators imply the existence of unbounded quasi-diagonal operators and exploring their implications for the structure of these spaces.

## Contribution

It establishes a link between unbounded operators and quasi-diagonal operators in l-K"othe spaces, providing new insights into their structure and operator boundedness.

## Key findings

- Existence of unbounded operators implies existence of unbounded quasi-diagonal operators.
- Conditions under which all continuous operators are bounded are characterized.
- Unbounded operators lead to the presence of a common basic subspace under certain conditions.

## 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-K\"{o}the spaces, then there exists a continuous unbounded quasi-diagonal operator between them. Using this result, we study in terms of corresponding K\"{o}the matrices when every continuous linear operator between l-K\"{o}the spaces is bounded. As an application, we observe that the existence of an unbounded operator between l-K\"{o}the spaces, under a splitting condition, causes the existence of a common basic subspace.