  For each $S \in L(E)$ (with $E$ a Banach space) the operator $R(S) \in
L(E^{**}/E)$ is defined by $R(S)(x^{**}+E) = S^{**}x^{**}+E$ \quad ($x^{**}\in
E^{**}$). We study mapping properties of the correspondence $S\to R(S),$ which
provides a representation $R$ of the weak Calkin algebra $L(E)/W(E)$ (here
$W(E)$ denotes the weakly compact operators on $E$). Our results display
strongly varying behaviour of $R.$ For instance, there are no non--zero compact
operators in Im$(R)$ in the case of $L^1$ and $C(0,1),$ but $R(L(E)/W(E))$
identifies isometrically with the class of lattice regular operators on
$\ell^2$ for $E=\ell^2(J)$ (here $J$ is the James' space). Accordingly, there
is an operator $T \in L(\ell^2(J))$ such that $R(T)$ is invertible but $T$
fails to be invertible modulo $W(\ell^2(J)).$
