Ideal structures in vector-valued polynomial spaces

Abstract

Note: An incorrect version of this article was posted from August 31, 2016, through September 7, 2016. The PDF is now correct.

This paper is concerned with the study of geometric structures in spaces of polynomials. More precisely, we discuss for $E$ and $F$ Banach spaces, whether the class of $n$-homogeneous polynomials, ${\mathcal{P}}_w{(}^{n}E,F)$, which are weakly continuous on bounded sets, is an HB-subspace or an $M(1,C)$-ideal in the space of continuous $n$-homogeneous polynomials, $\mathcal{P}{(}^{n}E,F)$. We establish sufficient conditions under which the problem can be positively solved. Some examples are given. We also study when some ideal structures pass from ${\mathcal{P}}_w{(}^{n}E,F)$ as an ideal in $\mathcal{P}{(}^{n}E,F)$ to the range space $F$ as an ideal in its bidual $F^{\ast \ast }$.