Representing non-weakly compact operators

Abstract

For each $S\in L(E)$ (with $E$ a Banach space) the operator $R(S)\in L(E^{\ast \ast }/E)$ is defined by $R(S)(x^{\ast \ast }+E)=S^{\ast \ast }{x}^{\ast \ast }+E$ \quad ($x^{\ast \ast }\in E^{\ast \ast }$). 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 $W({\ell }^2(J)).$