On quantitative self-duality of super weakly compact operators

It is well known that a bounded linear operator T between Banach spaces X and Y is super weakly compact if and only if so is its dual $T^{⁎}$. We study the quantitative version of this implication. The paper contains a counterexample showing that the super weak essential norms of a bounded linear operator and its dual are not equivalent. In detail, we construct a sequence of bounded linear operators $T_n:X\to Y$ so that the quotient norm ${\Vert T_n\Vert }_S$ induced by $L(X,Y)/S(X,Y)$ is not equivalent to ${\Vert T_n^{⁎}\Vert }_S$ induced by $L(Y^{⁎},X^{⁎})/S(Y^{⁎},X^{⁎})$. Above $L(X,Y)$ and $S(X,Y)$ stand for the collections of bounded linear operators and super weakly compact operators between X and Y, respectively. Our counterexample is derived from the Johnson-Lindenstrauss space.