Complemented subspaces of polynomial 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.