Orthogonality for Biadjoints of Operators

Abstract

Let X, Y be non-reflexive Banach spaces. Let ${\mathcal L}(X,Y)$ be the space of bounded operators from X to Y. For $T \in {\mathcal L}(X,Y)$, and a closed subspace $Z \subset Y$, this paper deals with the question, if $T \perp {\mathcal L}(X,Z)$ in the sense of Birkhoff-James, when is $T^{**} \perp {\mathcal L}(X^{**},Z^{\bot \bot })$? If $Z \subset Y$ is a subspace of finite codimension which is the kernel of projection of norm one, we show that this is always the case. Moreover in this case, there is an extreme point $\Lambda$ of the unit ball of the bidual $X^{**}$ such that $\Vert T^{**}\Vert = \Vert T^{**}(\Lambda )\Vert$ and $T^{**}(\Lambda ) \perp Z^{\bot \bot }$.