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

Abstract

We give geometrical characterization of Banach spaces W such that every representable operator from L 1( μ) into W admits a best approximation in the space of compact operators. This is the case if W = l 1( 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 l p , 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 L 1( 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 = l p , 1 < p < ∞, then every operator in l[V,C(Ω)] has a nearest point in K n[V,C(Ω)] .