Denseness results and Bollobás-type theorems for range strongly exposing operators

We present a new class of bounded linear operators on Banach spaces called Range strongly exposing operators (RSE, in short), which form a natural intermediate class between classical norm-attaining operators and Bourgain's absolutely strongly exposing operators (1977). In the first part of the talk, several foundational results on the denseness of norm-attaining operators are extended to this new setting. In particular, we improve some classical results by Uhl (1976) and Schachermayer (1983), and get some analogous to Acosta (1999). In the second part, we address Bollobás-type theorems for RSE operators, that is, results that allow us approximating simultaneously an operator and a point where the norm is almost attained, and we show that this property provides new characterisations of uniform convexity and complex-uniform convexity. This talk is based on joint works with Geunsu Choi, Audrey Fovelle, Mingu Jung and Miguel Martín.