Operator ideals on non-commutative function spaces

Abstract

Suppose $X$ and $Y$ are Banach spaces, and $\mathcal{I}$, $\mathcal{J}$ are operator ideals (for instance, the ideals of strictly singular, weakly compact, or compact operators). Under what conditions does the inclusion $\mathcal{I}(X,Y)\subset \mathcal{J}(X,Y)$, or the equality $\mathcal{I}(X,Y)=\mathcal{J}(X,Y)$, hold? We examine this question when $\mathcal{I},\mathcal{J}$ are the ideals of Dunford-Pettis, strictly (co)singular, finitely strictly singular, inessential, or (weakly) compact operators, while $X$ and $Y$ are non-commutative function spaces. Since such spaces are ordered, we also address the same questions for positive parts of such ideals.