Approximation properties defined by spaces of operators and approximability in operator topologies

Abstract

We develop a unified approach to characterize approximation properties defined by spaces of operators. Our main result describes them in terms of the approximability of weak*-weak continuous operators. In particular, we prove that if $\mathcal{A}$ and~$\mathcal{B}$ are operator ideals satisfying $\mathcal{A}\circ\mathcal{B}^{*}\subset\mathcal{K}$, then the $\mathcal{A}(X,X)$-approximation property of a Banach space $X$ is equivalent to the following “metric” condition: for every Banach space $Y$ and for every operator $T\in\mathcal{B}^{*}(Y,X)$, there exists a net $(S_{\alpha})\subset\mathcal{A}(X,X)$ such that $\sup_\alpha\|S_{\alpha}T\| \leq\|T\|$ and $T^{*}S_{\alpha}^{*}\rightarrow T^{*}$ in the strong operator topology on $\mathcal{L}(X^{\ast},Y^{\ast})$. As application, approximation properties of dual spaces and weak metric approximation properties are studied.