Operators Birkhoff–James Orthogonal to Spaces of Operators. (27th July 2021) - Rao, T. S. S. R. K.
Abstract: 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 x 0 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 L 1 -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 x 0 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.