Given an operator ideal $\mathfrak{I}$, $\mathfrak{I}^\text{dual}$ is the class of all operators $A:X\to Y$ between Banach spaces $X$ and $Y$ such that $A^*\in \mathfrak{I}$. Given an operator ideal $\mathfrak{I}$, it is often of interest to know when there is another ideal $\mathfrak{J}$ of independent interest such that either $\mathfrak{I}^\text{dual}=\mathfrak{J}$ or $\mathfrak{I}=\mathfrak{J}^\text{dual}$.

Natural examples are the compact and weakly compact operators, which are their own dual ideals. Another interesting example is that an operator $A:X\to Y$ is Asplund (or decomposing) if and only if $A^*$ has the Radon-Nikodym property. Therefore if $\mathfrak{P}$ denotes the ideal of Radon-Nikodym operators and $\mathfrak{D}$ denotes the ideal of Asplund operators, $\mathfrak{P}^\text{dual}=\mathfrak{D}$.

Let $\mathfrak{V}$ denote the class of completely continuous operators and let $\mathfrak{DP}$ denote the ideal of operators $B:X\to Y$ such that for any Banach space $Z$ and any weakly compact operator $A:Y\to Z$, $AB$ is completely continuous.

Is it known whether $\mathfrak{V}^\text{dual}$ or $\mathfrak{DP}^\text{dual}$ is another ideal of independent interest? Or whether there exists another ideal $\mathfrak{I}$ of independent interest such that either $\mathfrak{I}^\text{dual}= \mathfrak{V}$ or $\mathfrak{I}^\text{dual}= \mathfrak{DP}$?