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 operatorsare of finite rank exhaustively. Therefore, it is an area ofinterest in the concept of denseness of norm achieving map-pings to determine whether every finite rank operator betweenBanach spaces can be estimated by those that achieve theirnorms. Hence, it has not been shown whether the set of normachieving finite rank operators is dense in the whole space ofmappings of finite rank. We analyzed this properties partic-ulary norm-attainability. We showed that rank one mappingsachieve their norms on certain Banach lattices. Since finiterank operators are obtained by summing rank one mappings,our results seeks to clarify the behavior of such operators espe-cially in relation to norm-attainability and operator structurewith regard to Bishop-Phelps-Bollobas property.