Some new results on the class of AM-compact operators (Q708767)

An operator between Banach lattices is called \textit{AM-compact} if the image of each order interval \([-x,x]\), \(x\geq 0\), is norm relatively compact. In general, if \(T\) is AM-compact, its adjoint may not be AM-compact. Results relating the AM-compactness property of an operator to the AM-compactness property of its adjoint are provided. These results extend work of \textit{B. Aqzzouz}, \textit{R. Nouira} and \textit{L. Zraoula} [Can. Math. Bull. 51, No. 1, 15--20 (2008; Zbl 1149.47030)]. An example of the type of results obtained: Given \(E\) Dedekind \(\sigma\)-complete, then the property that each regular operator \(T:E\rightarrow E\) is AM-compact if and only if its adjoint is AM-compact is equivalent to both \(E\) and its dual having order continuous norms.