We explore the relation between the orthogonality of bounded linear operators in the space of operators and that of elements in the ground space. To be precise, we study if $ T, A \in \mathbb{ L}(\mathbb{ X}, \mathbb{ Y}) $ satisfy $ T \bot_B A,$ then whether there exists $ x \in \mathbb{ X} $ such that $ Tx\bot_B Ax$ with $ \|x\| =1, \|Tx\| = \|T\|$, where $\mathbb{ X}, \mathbb{ Y} $ are normed linear spaces. In this context, we introduce the notion of Property $ P_n $ for a Banach space and illustrate its connection with orthogonality of a bounded linear operator between Banach spaces. We further study Property $ P_n $ for various polyhedral Banach spaces.
Categories: math.FA