Operators Birkhoff–James Orthogonal to Spaces of Operators.

Let L (X , Y) be the space of bounded linear operators between real Banach spaces X, Y. For a closed subspace Z ⊂ Y , we partially solve the operator version of Birkhoff–James orthogonality problem, if T ∈ L (X , Y) is orthogonal to L (X , Z) , when does there exist a unit vector x₀ such that | | T (x 0) | | = | | T | | and T (x 0) is orthogonal to Z? In order to achieve this we first develop a compact optimization for a Y-valued compact operator T, via a minimax formula for d (T , L (X , Z)) in terms of point-wise best approximations, that links local optimization and global optimization. Our result gives an operator version of a classical minimax formula of Light and Cheney, proved for continuous vector-valued functions. For any separable reflexive Banach space X and for Z ⊂ Y is a L¹-predual space as well as a M-ideal in Y, we show that if T ∈ K (X , Y) is orthogonal to L (X , Z) , then there is a unit vector x₀ with | | T (x 0) | | = | | T | | and T (x 0) is orthogonal to Z. In the general case we also give some local conditions on T, when this can be achieved. [ABSTRACT FROM AUTHOR]