A unified approach to approximation properties and ideals
Volume 238 / 2017
Abstract
We use a unified argument to obtain relationships between approximation properties and ideals in spaces of some operators. We prove that a Banach space $X$ (respectively, the dual space $X^*$ of $X$) has the metric approximation property if and only if for every Banach space $Y$ and every operator $T$ from $Y$ to $X$ (respectively, $T$ from $X$ to $Y$), there exists a \begin{gather*} \varPhi\in \mathcal{HB}(\mathcal{F}(X)T, \, \textrm{span}(\mathcal{F}(X)T\cup\{T\}))\\ (\textrm{respectively,} \ \varPhi\in \mathcal{HB}(T\mathcal{F}(X), \, \textrm{span} (T\mathcal{F}(X)\cup\{T\}))) \end{gather*} such that $$ \varPhi(x^{*}\otimes y)(R)=x^{*}(Ry) \ (\textrm{respectively,} \ \varPhi(x^{**}\otimes y^{*})(R)=x^{**}(R^{\rm a}y^*)) $$ for every $x^{*}\in X^{*}$ and $y \in Y$ (respectively, $x^{**}\in X^{**}$ and $y^* \in Y^*$), and every $R \in \textrm{span} (\mathcal{F}(X)T\cup\{T\})$ (respectively, $R \in \textrm{span}(T\mathcal{F}(X)\cup\{T\}$)).
