Remarks on bounded operators in $\ell$-K\"othe spaces

Abstract

For locally convex spaces $X$ and $Y$, the continuous linear map $T:X \to Y$ is said to be bounded if it maps zero neighborhoods of $X$ into bounded sets of $Y$. We denote $(X,Y) \in \mathcal{B}$ when every operator between $X$ and $Y$ is bounded. For a Banach space $\ell$ with a monotone norm $\|\cdot\|$ in which the canonical system $(e_n)$ forms an unconditional basis, we consider $\ell$-K\"othe spaces as a generalization of usual K\"othe spaces. In this note, we characterize $\ell$-K\"othe spaces $\ell(a_{pn})$ and $\ell(b_{sm})$ such that $(\ell(a_{pn}), \ell(b_{sm})) \in \mathcal{B}$. A pair $(X,Y)$ is said to have the bounded factorization property, and denoted $(X,Y) \in \mathcal{BF}$, if each linear continuous operator $T : X \to X$ that factors over $Y$ is bounded. We also prove that injective tensor products of some classical K\"othe spaces have bounded factorization property.