Bollobás-type theorems for range strongly exposing operators

Helena Del Río

Paper

Bollobás-type theorems for range strongly exposing operators

Abstract

We study Bollobás-type theorems for range strongly exposing operators. When such a theorem holds for operators from a Banach space

into another Banach space

, we say that the pair

satisfies the Bishop-Phelps-Bollobás 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

and

. In particular, we show that

satisfies the BPBp-RSE if and only if

is uniformly convex, and that

satisfy the BPBp-RSE if and only if

-uniformly convex. We also highlight differences between the real and complex cases, showing that there exist pairs

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.