Ideals of operators, approximability in the strong operator topology, and the approximation property
The authors give characterisations of the approximation property of a Banach space \(X\) or its dual in terms of pointwise approximability of weakly compact operators by finite rank operators. One of their results says that \(X^*\) has the approximation property if and only if for every Banach space \(Z\) and every weakly compact operator \(T: Z\to X\) there is a net of finite rank operators \(T_\alpha : Z\to X\) of norm \(\| T_\alpha \| \leq \| T\| \) converging to \(T\) pointwise. They also give characterisations of when, for example, for all \(Z\), the space of compact operators \(K(Z,X)\) is an ideal in the space of weakly compact operators \(W(Z,X)\) or \(W(Z,X^{**})\) in the same spirit. (A subspace of a Banach space is called an ideal if its annihilator is the kernel of a norm-\(1\) projection in the dual.) These considerations permit the authors to exhibit a space of compact operators \(K(Z,X)\) that is not an \(M\)-ideal in \(K(Z,X^{**})\) although \(X\) is an \(M\)-ideal in \(X^{**}\). This surprising example solves a problem left open in the reviewer's paper [Math.\ Scand.\ 61, No. 1, 149--164 (1987; Zbl 0619.46062)] where this property of \(K(Z,X)\) is shown to hold if \(X^{**}\) has the approximation property.
scientific article; zbMATH DE number 2112900
scientific article