Remarks on bounded operators in $\ell$-Köthe 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 $\Vert \cdot \Vert$ in which the canonical system $(e_n)$ forms an unconditional basis, we consider $\ell$-Köthe spaces as a generalization of usual Köthe spaces. In this note, we characterize $\ell$-Köthe 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 $(X,Y)\in \mathcal{B}\mathcal{F}$, 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 Köthe spaces have bounded factorization property.

Withdrawn due to an error in Theorem 2.1