On Certain properties of Finite Rank Operators and Bishop-Phelps-Bollobas Criterion in Banach Spaces

Abstract

For bounded linear operators, it is observed that, Bishop-Phelps-Bollobas criterion has not been studied when operators are of finite rank exhaustively. Therefore, it is an area of interest in the concept of denseness of norm achieving mappings to determine whether every finite rank operator between Banach spaces can be estimated by those that achieve their norms. Hence, it has not been shown whether the set of norm achieving finite rank operators is dense in the whole space of mappings of finite rank. We analyzed this properties particulary norm-attainability. We showed that rank one mappings achieve their norms on certain Banach lattices. Since finite rank operators are obtained by summing rank one mappings, our results seeks to clarify the behavior of such operators especially in relation to norm-attainability and operator structure with regard to Bishop-Phelps-Bollobas property.