Duality of the weak essential norm (Q2701639)

Let \(E\) and \(F\) be Banach spaces, and let \({\mathcal L}(E,F)\) be the space of all bounded linear operators from \(E\) into \(F\). Let \(W(E,F)\) denote a (closed) subspace of \({\mathcal L}(E,F)\) composed of all weakly compact operators. The weak essential norm is the quotient norm \(\|S\|_w=\text{dist}(S,W(E,F))\), \(S\in{\mathcal L}(E,F)\). The classical result due to V.R.~Gantmacher implies that \(\|S^\ast\|_w\leq\|S\|_w\) for \(S\in {\mathcal L}(E,F)\). It is a natural problem whether there is a converse estimate, i.e. if there is a constant \(c>0\) such that NEWLINE\[NEWLINE c\|S\|_w\leq\|S^\ast\|_w,\quad (S\in{\mathcal L}(E,F))? NEWLINE\]NEWLINE The author gives a counterexample to this question. The construction is based on a Banach space introduced by \textit{W. B. Johnson} and \textit{J. Lindenstrauss} in their study of weakly compactly generated Banach spaces [Isr. J. Math. 17, 219-230 (1974; Zbl 0306.46021)].