Best approximation in spaces of compact operators.

Let K (X , Y) be the space of compact operators. For a proximinal subspace Z ⊂ Y , this paper deals with the question, when does every Y -valued compact operator admit a Z -valued compact best approximation? For any reflexive Banach space X and for a L 1 -predual space Y , if Z ⊂ Y is a strongly proximinal subspace of finite codimension, we show that K (X , Z) is a proximinal subspace of K (X , Y) under an additional condition on the position of K (X , Z). When Y is a c 0 -direct sum of finite dimensional spaces we achieve a strong transitivity result by showing that for any proximinal subspace of finite codimension Z ⊂ Y , every Y -valued bounded operator admits a best Z -valued compact approximation. [ABSTRACT FROM AUTHOR]