Complemented subspaces of polynomial ideals

TL;DR

This paper investigates conditions under which polynomial ideals are not complemented in larger spaces, showing that containing an isomorphic copy of c0 prevents such complementarity, generalizing previous results.

Contribution

It establishes new non-complementarity results for polynomial ideals containing c0, extending prior work to broader classes of operator ideals.

Findings

Polynomial ideals containing c0 are not complemented in larger polynomial spaces.

Results apply to all closed operator ideals within the compact operators.

Generalizes previous non-complementarity results for specific operator ideals.

Abstract

Given the polynomial ideal $\mathcal{J}\circ\mathcal{P} (^{n}E; F)$, we prove that if $\mathcal{J}\circ\mathcal{P} (^{n}E; F)$ contains an isomorphic copy of $c_{0}$, then $\mathcal{J}\circ\mathcal{P} (^{n}E; F)$ is not complemented in $\mathcal{P} (^{n}E; F)$ for every closed operator ideal $\mathcal{J}\subset \mathcal{L}_{K}$ and every $n\in\mathbb{N}$. Likewise we show that if $\widehat{(\mathcal{J}\circ\mathcal{L})^{fac}}(^{n}E;F)$ contains an isomorphic copy of $c_{0}$, then $\widehat{(\mathcal{J}\circ\mathcal{L})^{fac}}(^{n}E;F)$ is not complemented in $\mathcal{P}(^{n}E; F)$ for every closed operator ideal $\mathcal{J}\subset \mathcal{L}_{K}$ and every $n>1$. When $\mathcal{J}=\mathcal{L}_{K}$, these results generalizes results of several authors \cite{LEW},\cite{EM},\cite{KALTON},\cite{IOANA},\cite{SERGIO}, among others.