Bounded Operators to $\ell$-K\"othe Spaces

Abstract

For Fr{\'e}chet spaces E and F we write (E,F) \in {B} if every continuous linear operator from E to F is bounded. Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We obtain a necessary and sufficient condition for (E,F) \in {B} when F = \lambda^{l}(B). We say that a triple (E,F,G) has the bounded factorization property and write (E,F,G) \in {BF} if each continuous linear operator T : E \longrightarrow G that factors over F is bounded. We extend some results in \cite{Ter03} to l-K\"{o}the spaces and obtain a sufficient condition for (E,\lambda^{l_1}(A) \hat{otimes}_{pi} \lambda^{l_2}(B), \lambda^{l_3}(C)) \in {BF} when \lambda^{l_1}(A) and \lambda^{l_2}(B) are nuclear.