linear space. An element $ x \in S_{\mathbb{X}} $ is said to be a smooth point if there exists a unique supporting hyperplane to $B_{\mathbb{X}}$ at $x.$ The characterization of smooth points obtained by James [@J] has been used in our study, which states that $ x \in S_{\mathbb{X}} $ is a smooth point if and only if $ x \bot_B y $ and $ x \bot_B z$ implies $ x \bot_B {(y + z)}.$

Main results.
=============

Let us begin by giving a complete characterization of Birkhoff-James orthogonality of compact linear operators defined on a reflexive Banach space. This extends Theorem $ 2.2 $ of [@S].

\[theorem:compact\] Let $\mathbb{X}$ be a reflexive Banach space and $\mathbb{Y}$ be any normed linear space. Then for any $T, A\in \mathbb{K}(\mathbb{X}, \mathbb{Y})$, $T\bot_B A$ if and only if there exists $x, y\in M_T$ such that $Ax\in (Tx)^+$ and $Ay\in (Ty)^-$.

Let us first prove the easier sufficient part.\
Since $Ax\in (Tx)^+,~\|T+\lambda A\|\geq \|Tx+ \lambda Ax\|\geq \|Tx\|=\|T\|$ for all $\lambda \geq 0$. Similarly $Ay\in (Ty)^-$ implies that $\|T+\lambda A\|\geq \|Ty+ \lambda Ay\|\geq \|Ty\|=\|T\|$ for all $\lambda \leq 0$. This completes the proof of the sufficient part.\
Let us now prove the necessary part.\
Since 