# Norm attaining operators of finite rank

## TL;DR

This paper investigates conditions under which Banach spaces admit norm attaining operators of finite rank, providing characterizations, density results, and potential counterexamples related to the Bishop-Phelps theorem.

## Contribution

It offers new sufficient conditions for the existence and density of finite rank norm attaining operators and characterizes these properties in Hilbert spaces.

## Key findings

- Existence of non-trivial cones of norm attaining functionals implies finite rank norm attaining operators.
- Density of finite rank norm attaining operators holds under certain conditions.
- A candidate counterexample to the complex Bishop-Phelps theorem on c0 is proposed.

## Abstract

We provide sufficient conditions on a Banach space $X$ in order that there exist norm attaining operators of rank at least two from $X$ into any Banach space of dimension at least two. For example, a rather weak such condition is the existence of a non-trivial cone consisting of norm attaining functionals on $X$. We go on to discuss density of norm attaining operators of finite rank among all operators of finite rank, which holds for instance when there is a dense linear subspace consisting of norm attaining functionals on $X$. In particular, we consider the case of Hilbert space valued operators where we obtain a complete characterization of these properties. In the final section we offer a candidate for a counterexample to the complex Bishop-Phelps theorem on $c_0$, the first such counterexample on a certain complex Banach space being due to V. Lomonosov.