$M$-structures in vector-valued polynomial spaces

TL;DR

This paper investigates the conditions under which classes of weakly continuous polynomials form $M$-ideals in spaces of continuous polynomials, extending the concept to vector-valued polynomial spaces and analyzing specific Banach space cases.

Contribution

It introduces criteria for $M$-structure existence in vector-valued polynomial spaces and extends property $(M)$ to this setting, providing new insights and results.

Findings

$M$-structures occur only for finite $n$ in certain cases

Established conditions for positive and negative $M$-structure results

Extended property $(M)$ to vector-valued polynomial spaces

Abstract

This paper is concerned with the study of $M$-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 $M$-ideal in the space of continuous $n$-homogeneous polynomials $\mathcal P(^n E, F)$. We show that there is some hope for this to happen only for a finite range of values of $n$. We establish sufficient conditions under which the problem has positive and negative answers and use the obtained results to study the particular cases when $E=\ell_p$ and $F=\ell_q$ or $F$ is a Lorentz sequence space $d(w,q)$. We extend to our setting the notion of property $(M)$ introduced by Kalton which allows us to lift $M$-structures from the linear to the vector-valued polynomial context. Also, when $\mathcal P_w(^n E, F)$ is an $M$-ideal in $\mathcal P(^n E, F)$ we prove a Bishop-Phelps type result for vector-valued polynomials and relate norm-attaining polynomials with farthest points and remotal sets.