Some Remarks on Orthogonality of Bounded Linear Operators

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 is an element of L(X, Y) satisfy T perpendicular to(B)A, then whether there exists x is an element of X such that Tx perpendicular to(B)Ax with parallel to x parallel to = 1, parallel to Tx parallel to = parallel to T parallel to, where X, 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.