Birkhoff-James orthogonality and smoothness of bounded linear operators

TL;DR

This paper establishes new conditions for the smoothness of bounded linear operators on Banach and Hilbert spaces, linking Birkhoff-James orthogonality with operator properties and norm attainment.

Contribution

It introduces sufficient conditions for operator smoothness and orthogonality characterizations, extending previous results to Banach and Hilbert space operators.

Findings

Characterization of Birkhoff-James orthogonality for operators on Banach spaces.

Conditions for an operator to be a smooth point in operator spaces.

Description of the norm attaining set for operators on Hilbert spaces.

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.