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.
Categories: math.FA