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 $forsome$ 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}${y \in C} |Ty| < |T|$forallclosedsubsets$C$of$S{$\mathbb{X}$}$with$d(\pm x,C) > 0.$ForoperatorsonaHilbertspace$ $\mathbb{H}$$weshowthat$ T \bot_{B} A \Leftrightarrow Tx \bot_{B} Ax $forsome$ x \in S_{$\mathbb{H}$}$with$ |Tx| = |T| $ifandonlyifthenormattainingset$M_T = { x \in S_{$\mathbb{H}$} : |Tx| = |T| } = S_{H_0}$ for some finite dimensional subspace $H_0$ and $\mathrm{\parallel }T{\mathrm{\parallel }}_{{H_o}^{\mathrm{\perp }}}<\mathrm{\parallel }T\mathrm{\parallel }\mathrm{.}$ We also characterize smoothness of compact operators on normed spaces and bounded linear operators on Hilbert spaces.