Abstract We study the representation of non-weakly compact operators between AL -spaces. In this setting, we show that every operator admits a best approximant in the ideal of weakly compact operators. Using duality arguments, we extend this result to operators between C (L) -spaces where L is extremally disconnected. We also characterize the weak essential norm for operators between AL -spaces in terms of factorizations of the identity on ₁. As a consequence, we deduce that the weak Calkin algebra B (E) /W (E) admits a unique algebra norm for every AL -space E. By duality, similar results are obtained for C (K) -spaces. In particular, we prove that for operators T: L_0, 1 L_0, 1 the weak essential norm, the residuum norm, and the De Blasi measure of weak compactness coincide, answering a question of González, Saksman and Tylli.
Acuaviva et al. (Fri,) studied this question.
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