our notations and terminologies.\
In this paper, letters $ \mathbb{X}, \mathbb{Y} $ denote normed linear spaces. Throughout the paper, we will be working with real normed linear spaces. Let $B_{\mathbb{X}} = \{x \in \mathbb{X} \colon \|x\| \leq 1\}$ and $S_{\mathbb{X}} = \{x \in \mathbb{X} \colon \|x\|=1\}$ be the unit ball and the unit sphere of $\mathbb{X}$ respectively. Let $\mathbb{B}(\mathbb{X}, \mathbb{Y}) (\mathbb{K}(\mathbb{X}, \mathbb{Y})) $ denote the space of all bounded (compact) linear operators from $ \mathbb{X} $ to $ \mathbb{Y}. $ For $x,y \in \mathbb{X}$, $x$ is said to be *orthogonal to $y$ in the sense of Birkhoff-James* [@B], written as $x \perp_B y$, if $\|x\| \leq \|x+\lambda y\|$ for all $\lambda \in \mathbb{R}$. Likewise for $T,A \in \mathbb{B}(\mathbb{X}, \mathbb{Y})$, $T$ is said to be orthogonal to $A$ in the sense of Birkhoff-James, written as $T \perp_B A$, if $\|T\| \leq \|T + \lambda A\|$ for all $\lambda \in  \mathbb{R}$. It is easy to observe that in inner product spaces, $x \perp_B y$ is equivalent to the usual inner product orthogonality $x \perp y$.

In a Hilbert space $\mathbb{H}$, Bhatia and Šemrl [@BS] and Paul [@P] independently proved that $T \perp_B A$ if and only if there exists $\{x_n\} \subset S_{\mathbb{H}}$ 