On the norm attainment set of a bounded linear operator

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 nonzero bounded linear operator to attain norm at a particular point of the unit sphere. We prove four 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 lp2(p∈N∖{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.