Q-Reflexive Banach spaces

Abstract

Let $E$ be a Banach space and, for any positive integer $n$, let $\mathcal{P}{(}^{n}E)$ denote the Banach space of continuous $n$-homogeneous polynomials on $E$. Davie and Gamelin showed that the natural extension mapping from $\mathcal{P}{(}^{n}E)$ to $\mathcal{P}{(}^n{E}^{\ast \ast })$ is an isometry into the latter space. Here, we investigate when there is a natural isomorphism between $\mathcal{P}{(}^{n}E{)}^{\ast \ast }$ and $\mathcal{P}{(}^n{E}^{\ast \ast })$. Among other things, we show that if $E$ satisfies: \break (a) no spreading model built on a normalised weakly null sequence has a lower $q$-estimate for any $q<\infty ,$ (b) $E^{\ast }$ has RNP, and (c) $E^{\ast }$ has the approximation property, then $\mathcal{P}{(}^{n}E)$ has RNP for every $n$. Moreover, if $E$ satisfies (a) and is such that $E^{\ast \ast }$ has both the RNP and the approximation property, then $\mathcal{P}{(}^{n}E{)}^{\ast \ast }$ and $\mathcal{P}{(}^n{E}^{\ast \ast })$ are isomorphic for every $n$. We also exhibit a quasi-reflexive Banach space $E$ for which $\mathcal{P}{(}^{n}E{)}^{\ast \ast }$ and $\mathcal{P}{(}^n{E}^{\ast \ast })$ are isomorphic for every $n$. Related work has been done recently by (i) M. Gonzalez, (ii) M. Valdivia, and (iii) J. Jaramillo, A. Prieto, and I. Zalduendo.

References in corpus (1)

• Unconditionally converging polynomials on Banach spaces