A bounded approximation of weakly compact operators (Q1941340)

A Banach space \(X\) has the approximation property if the identity operator \(\mathrm{id}_X\) can be approximated by finite rank operators in the topology \(\tau\) of uniform convergence on compact subsets of \(X\). \(X\) has the bounded approximation property if this can be done by finite rank operators in some ball. The goal of this paper is to study the approximation of other bounded operators in \(\mathcal{L}(X)\) by finite rank operators. The main result of the paper is that, if \(T\) is weakly compact in \(\mathcal{L}(X)\) and \(T\) can be approximated by finite rank operators in the topology \(\tau\), then \(T^2\) can be approximated by finite rank operators with norm bounded by \(\|T\|^2\). Variants of the weak bounded approximation property for \(T\) in \(\mathcal{L}(X)\) are also studied.