On the norm attainment set of a bounded linear operator

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:457

Article

Sain, Debmalya¹ 

[1] Indian Inst Sci, Dept Math, Bengaluru 560012, Karnataka, India

关键词: Linear operator;    Norm attainment;    Birkhoff-James orthogonality;    Smooth Banach space;   

DOI  :  10.1016/j.jmaa.2017.07.070

来源: Elsevier

【 摘 要 】

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 l(p)(2)(p is an element of N backslash {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. (c) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free