Generalized Kӧthe $p$-dual spaces

Abstract

Let us consider a Banach function space $X$. The Kӧthe dual space can be characterized as the space of multipliers from $X$ to $L^1$. We extend this characterization to the space of multipliers from $X$ to $L^p$ in order to define the Kӧthe $p$-dual space of $X$. We analyze the properties of this space so as to use it as a tool for studying $p$-th power factorable operators. In particular, we compute $q$-concavity for these spaces and type and cotype when $X$ is an AM-space. As main applications, we give a characterization for Hilbert Banach function spaces, as well as a factorization for $p$-th power factorable operators through an $L^{p,\infty}$-space.