Ideal structures in vector-valued polynomial spaces

TL;DR

This paper investigates the geometric ideal structures within spaces of vector-valued polynomials, establishing conditions for when certain polynomial classes form HB-subspaces or $M(1,C)$-ideals, with implications for the structure of Banach spaces.

Contribution

It provides new sufficient conditions for polynomial classes to be HB-subspaces or $M(1,C)$-ideals in vector-valued polynomial spaces, extending understanding of their geometric structure.

Findings

Identified conditions for $\\mathcal P_w(^n E, F)$ to be an HB-subspace or $M(1,C)$-ideal.

Examples illustrating when ideal structures pass to the bidual of the range space.

Extended the theory of geometric structures in polynomial spaces.

Abstract

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 weakly continuous on bounded sets $n$-homogeneous polynomials, $\mathcal P_w(^n E, F)$, 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 $F^{**}$.