Bollob\'as-type theorems for range strongly exposing operators

Abstract

We study Bollob\'as-type theorems for range strongly exposing operators. When such a theorem holds for operators from a Banach space $X$ into another Banach space $Y$, we say that the pair $(X,Y)$ satisfies the Bishop-Phelps-Bollob\'as property for range strongly exposing operators (BPBp-RSE, for short). We provide new characterisations of uniform convexity and complex uniform convexity via the BPBp-RSE, including for pairs involving spaces such as $L_1(\mu), L_\infty(\mu)$ and $c_0$. In particular, we show that $(L_1(\mu), Y)$ satisfies the BPBp-RSE if and only if $Y$ is uniformly convex, and that $(L_\infty(\mu), Y)$ or $(c_0, Y)$ satisfy the BPBp-RSE if and only if $Y$ is $\mathbb{C}$-uniformly convex. We also highlight differences between the real and complex cases, showing that there exist pairs $(X, Y)$ for which the BPBp-RSE holds in the complex setting but fails for their respective underlying real spaces. Additionally, we consider various subspaces of operators, such as compact and finite-rank, and extend several results from the literature to this new setting. The paper concludes with a collection of open problems.