REPRESENTING NON-WEAKLY COMPACT-OPERATORS

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)).