Semicompact operators
Let \({\mathcal S}^ r(E,F)\) (E and F are Banach lattices with F Dedekind complete) be the space of all regular operators T from E to F such that \(| T|\) maps norm bounded sets into almost order bounded once. As a followup to [\textit{W. Arendt} and \textit{H. U. Schwarz}, Math. Nachrichten 131, 7-18 (1987; Zbl 0641.47044)] this paper gives answers to the following problems: (a) When is \({\mathcal S}^ r(E,F)\) equal to the space of all regular operators? (b) When does \({\mathcal K}^ r(E,F)={\mathcal S}^ r(E,F)\cap (E'\otimes F)^{\perp \perp}\) hold? (c) When is \(E'\otimes F\) dense in \({\mathcal S}^ r(E,F)\) with respect to the r-norm? (d) When does \({\mathcal S}^ r(E,F)\) contain \((E'\otimes F)^{\perp \perp}?\)
scientific article; zbMATH DE number 4165785