Birkhoff-James orthogonality and smoothness of bounded linear operators
Kallol Paul, Debmalya Sain, Puja Ghosh
Abstract
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.