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.
Categories: math.OA