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Representing non-weakly compact operators. (English) Zbl 0832.47039

Summary: For each \(S\in L(E)\) (with \(E\) a Banach space) the operator \(R(S)\in L(E^{**}/ E)\) is defined by \(R(S) (x^{**}+ E)= S^{**} x^{**}+ E\) \((x^{**}\in E^{**})\). 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 \(\text{Im} (R)\) 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 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 \(W(\ell^2 (J))\).

MSC:

47L10 Algebras of operators on Banach spaces and other topological linear spaces
47B07 Linear operators defined by compactness properties
46B28 Spaces of operators; tensor products; approximation properties
47A67 Representation theory of linear operators
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