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

TL;DR

This paper introduces the Bishop-Phelps-Bollobás property for range strongly exposing operators, characterizes uniform convexity via this property, and explores differences between real and complex cases in Banach space operator theory.

Contribution

It provides new characterizations of uniform convexity using the BPBp-RSE and extends the theory to subspaces of operators, highlighting differences between real and complex spaces.

Findings

$(L_1(), Y)$ satisfies BPBp-RSE iff $Y$ is uniformly convex.

$(L_(), Y)$ or $(c_0, Y)$ satisfy BPBp-RSE iff $Y$ is -uniformly convex.

Existence of pairs where BPBp-RSE holds complex but not in the real setting.

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.