Best approximation in spaces of compact operators (Q2041762)

In this paper, the author addressed an important question regarding best approximations in spaces of compact operators. For a proximinal subspace \(Z \subset Y\) of the space of compact operators \(\mathcal{K}(X,Y)\); the following question was taken up in the paper: 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 \(Y\), it is shown that if \(Z \subset Y\) is a strongly proximinal subspace of finite codimension, then \(\mathcal{K}(X,Z)\) is a proximinal subspace of \(\mathcal{K}(X,Y)\) under an additional condition on the position of \(\mathcal{K}(X,Z)\). Also, when \(Y\) is a \(c_0\)-direct sum of finite dimensional spaces, a strong transitivity result is achieved by showing that for any proximinal subspace of finite codimension \(Z \subset Y\) , every \(Y\)-valued bounded operator intermits a best \(Z\)-valued compact approximation. Indeed, this paper paves an innovative approach to study best approximations in spaces of compact operators.