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 $\mathfrak{I}^\text{dual}= \mathfrak{DP}$? 



Duals of ideals of operators between Banach spaces
Let $\mathcal{K}$, $\mathcal{W}$ and $\mathcal{C}$ denote the compact, weakly compact and completely continuous operators, respectively, and let $\mathcal{W}^{-1}\circ \mathcal{K}$ denote the operators $T:X\to Y$ such that $BT\in\mathcal{K}$ for each $Z$ and each $B\in\mathcal{W}(Y,Z)$. 


CLAIM: $\mathcal{C}^{dual} = \mathcal{W}^{-1}\circ \mathcal{K}$. 


Suppose $T\in \mathcal{C}^{dual}(X,Y)$. 
If $B\in\mathcal{W}(Y,Z)$, then $(BT)^\*=T^\*B^\* \in \mathcal{K}$ because $T^\*\in \mathcal{C}$ and $B^\*\in\mathcal{W}$, hence $BT\in \mathcal{K}$, and we conclude $T\in\mathcal{W}^{-1}\circ \mathcal{K}$. 


Suppose $T\in (\mathcal{W}^{-1}\circ \mathcal{K})(X,Y)$, and let $(g\_n)$ be a weakly null sequence in $Y^\*$. 
The operator $B:Y\to c\_0$ defined by $Bx =(g\_n(x))$ is weakly compact. Then $BT\in \mathcal{K}$. Since the unit vector basis $(e\_n)$ is weak$^\*$-null in $\ell\_1=c\_0^\*$, $(T^\*g\_n)= (T^\*B^\*e\_n)$ is norm null. Thus $T^\*\in \mathcal{C}$. 


