Approximation des bornés d'un espace Banach par des compacts et applications à l'approximation des opérateurs bornés

AbstractWe give geometrical characterization of Banach spaces W such that every representable operator from L1(μ) into W admits a best approximation in the space of compact operators. This is the case if W = l1(I), W = C(Ω), or W is a uniformely convex Banach space. In the dual situation we study the existence of best compact approximation for operators from a Banach space V into C(Ω). Such approximations exist if V is lp, 1 < p < ∞. We study also the existence of best approximation in the set of operators of a given finite dimensional range for representable operators from L1(μ) into a Banach space W. The problem is solved when there is a norm one linear projection from W″ onto W. As to operators with values in C(Ω), it is proved that if V = lp, 1 < p < ∞, then every operator in l[V,C(Ω)] has a nearest point in Kn[V,C(Ω)]