On the norm attainment set of a bounded linear operator

TL;DR

This paper investigates the properties of bounded linear operators on Banach spaces related to norm attainment, utilizing Birkhoff-James orthogonality to derive conditions, characterizations, and bounds relevant to operator behavior and space smoothness.

Contribution

It provides a necessary condition for norm attainment using Birkhoff-James orthogonality and characterizes smooth Banach spaces through this framework, also establishing bounds in specific spaces.

Findings

Necessary condition for norm attainment at a point on the unit sphere.

Characterization of smooth Banach spaces via operator norm attainment.

Upper bounds on the number of norm attainment points in l_p^2 spaces.

Abstract

In this paper we explore the properties of a bounded linear operator defined on a Banach space, in light of operator norm attainment. Using Birkhoff-James orthogonality techniques, we give a necessary condition for a bounded linear operator attaining norm at a particular point of the unit sphere. We prove a number of corollaries to establish the importance of our study. As part of our exploration, we also obtain a characterization of smooth Banach spaces in terms of operator norm attainment and Birkhoff-James orthogonality. Restricting our attention to $ l_{p}^{2} (p \in \mathbb{N}\setminus \{ 1 \})$ spaces, we obtain an upper bound for the number of points at which any linear operator, which is not a scalar multiple of an isometry, may attain norm.