For each S is an element of L(E) (with E a Banach space) the operator
R(S) is an element of L(E*/E) is defined by R(S)(x** + E) = S**x** E (x* is an element of E**). We study mapping properties of the corre
spondence S --> R(S), which provides a representation R of the weak Ca
lkin algebra L(E)/W(E) (here W(E) denotes the weakly compact operators
on E). Our results display strongly varying behaviour of R. For insta
nce, there are no non-zero compact operators in Im(R) in the case of L
(1) and C(O, 1), but R(L(E)/W(E)) identifies isometrically with the cl
ass of lattice regular operators on l(2) for E = l(2)(J) (here J is Ja
mes' space). Accordingly, there is an operator T is an element of L(l(
2)(J) such that R(T)is invertible but T fails to be invertible module
W(l(2)(J)).