Birkhoff-James orthogonality and smoothness of bounded linear operators

We present a sufficient condition for smoothness of bounded linear operators on Banach spaces for the first time. Let $T, A \in B(\mathbb{X}, \mathbb{Y}),$ where $\mathbb{X}$ is a real Banach space and $\mathbb{Y}$ is a real normed linear space. We find sufficient condition for $T \bot_{B} A \Leftrightarrow Tx \bot_{B} Ax$ for some $x \in S_{\mathbb{X}}$ with $\|Tx\| = \|T\|,$ and use it to show that $T$ is a smooth point in $B(\mathbb{X}, \mathbb{Y})$ if $T$ attains its norm at unique (upto muliplication by scalar) vector $x \in S_{\mathbb{X}},$ $Tx$ is a smooth point of $\mathbb{Y}$ and {\em sup}$_{y \in C} \|Ty\| < \|T\|$ for all closed subsets $C$ of $S_{\mathbb{X}}$ with $d(\pm x,C) > 0.$ For operators on a Hilbert space $\mathbb{H}$ we show that $T \bot_{B} A \Leftrightarrow Tx \bot_{B} Ax$ for some $x \in S_{\mathbb{H}}$ with $\|Tx\| = \|T\|$ if and only if the norm attaining set $M_T = \{ x \in S_{\mathbb{H}} : \|Tx\| = \|T\| \} = S_{H_0}$ for some finite dimensional subspace $H_0$ and $\|T\|_{{H_o}^{\bot}} < \|T\|.$ We also characterize smoothness of compact operators on normed spaces and bounded linear operators on Hilbert spaces.