On Birkhoff-James Orthogonality and Norm-Attainability of Operators in Banach Spaces

Abstract

Characterizing geometric properties in Banach spaces in terms of their mappings has been done for a long period of time however it remains a very difficult task to complete due to the complex underlying structures in the Banach spaces. Recently, of interest has been the norm-attainability and orthogonality aspects in Banach space setting in general. To char- acterize these properties, one requires a geometrical view of the problem and this brings into the picture the concept of Birkhoff-James orthogo- nality in order to solve the problem. The main objective of this study is to establish norm-attainability conditions of operators via Birkhoff-James orthogonality in Banach spaces. The specific objectives include to: Es- tablish Birkhoff-James orthogonality conditions for operators in Banach spaces; Determine norm-attainability of operators in Banach spaces via Birkhoff-James orthogonality and; Investigate the relationship between the set of norm-attainable vectors and the set of norm-attainable oper- ators via Birkhoff-James orthogonality in Banach spaces. The research methodology involved the use of known orthogonality criterion in normed spaces, technical approaches such as polar decomposition and tensor prod- ucts and some known inequalities such as triangle inequality and Cauchy- Schwarz inequality. The results show that operators are norm-attainable in Banach spaces via Birkhoff-James orthogonality. Moreover, there is a strong relationship between the set of norm-attainable operators and the set of norm-attainable vectors. The results of this study are useful in understanding the concept of orthogonal projections and has applications in optimization theory and convex analysis.