Birkhoff-James orthogonality and smoothness of bounded linear operators

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 $T{\perp }_{B}A\iff Tx{\perp }_{B}Ax$ for some $x\in S_{\mathbb{X}}$ with $\Vert Tx\Vert =\Vert T\Vert ,$ 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}${}_{y\in C}\Vert Ty\Vert <\Vert T\Vert$ 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 $T{\perp }_{B}A\iff Tx{\perp }_{B}Ax$ for some $x\in S_{\mathbb{H}}$ with $\Vert Tx\Vert =\Vert T\Vert$ if and only if the norm attaining set $M_T=\{x\in S_{\mathbb{H}}:\Vert Tx\Vert =\Vert T\Vert \}=S_{H_0}$ for some finite dimensional subspace $H_0$ and $\Vert T{\Vert }_{{H_o}^{\perp }}<\Vert T\Vert .$ We also characterize smoothness of compact operators on normed spaces and bounded linear operators on Hilbert spaces.

11 pages

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• Orthogonality and Numerical radius inequalities of operator matrices
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