Weakening of a local Bollobás type property and geometry of Banach spaces

This paper deals with a weaker form of the property so called \({\textbf {L}}_{o,o}\) for operators, which we call the property weak \({\textbf {L}}_{o,o}\) for operators. We characterize this property in terms of convergence of approximate norm attainment sets and prove that a pair of Banach spaces (X, Y) satisfies the property weak \({\textbf {L}}_{o,o}\) for compact operators if and only if X is reflexive. We further investigate the property weak \({\textbf {L}}_{o,o}\) for bilinear maps and obtain a connection of it with the property weak \({\textbf {L}}_{o,o}\) for operators. Importantly, we also characterize some geometric properties of Banach spaces with the help of convergence of approximate norm attainment sets.