  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.
