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.

arXiv admin note: substantial text overlap with arXiv:1612.01742

References in corpus (1)

• Approximation properties determined by operator ideals