Compactness and an approximation property related to an operator ideal

Abstract

For an operator ideal $\mathcal{A}$, we study the composition operator ideals $\mathcal{A}\circ \mathcal{K}$, $\mathcal{K}\circ \mathcal{A}$ and $\mathcal{K}\circ \mathcal{A}\circ \mathcal{K}$, where $\mathcal{K}$ is the ideal of compact operators. We introduce a notion of an $\mathcal{A}$-approximation property on a Banach space and characterise it in terms of the density of finite rank operators in $\mathcal{A}\circ \mathcal{K}$ and $\mathcal{K}\circ \mathcal{A}$. We propose the notions of ${\ell }_{\infty }$-extension and ${\ell }_1$-lifting properties for an operator ideal $\mathcal{A}$ and study $\mathcal{A}\circ \mathcal{K}$, ${\mathcal}\circ{\mathcal A}$ and the $\mathcal{A}$-approximation property where $\mathcal{A}$ is injective or surjective and/or with the ${\ell }_{\infty }$-extension or ${\ell }_1$-lifting property. In particular, we show that if $\mathcal{A}$ is an injective operator ideal with the ${\ell }_{\infty }$-extension property, then we have: (a) $X$ has the $\mathcal{A}$-approximation property if and only if $({\mathcal{A}}^{min}{)}^{inj}(Y,X)={\mathcal{A}}^{min}(Y,X)$, for all Banach spaces $Y$. (b) The dual space $X^{\ast }$ has the $\mathcal{A}$-approximation property if and only if $(({\mathcal{A}}^{dual}{)}^{min}{)}^{sur}(X,Y)=({\mathcal{A}}^{dual}{)}^{min}(X,Y)$, for all Banach spaces $Y$.}For an operator ideal $\mathcal{A}$, we study the composition operator ideals $\mathcal{A}\circ \mathcal{K}$,

23 pages