# Bishop-Phelps-Bollob\'as property for positive operators between classical Banach spaces

## TL;DR

This paper establishes the Bishop-Phelps-Bollobás property for positive operators between certain classical Banach spaces, specifically from $L_()$ to $L_1(
u)$ and from $c_0$ to $$, highlighting both positive results and limitations.

## Contribution

It proves the Bishop-Phelps-Bollobás property for positive operators on specific Banach space pairs and provides a counterexample for general Banach lattices.

## Key findings

- Positive operators from $L_()$ to $L_1(
u)$ have the property.
- The pair $(c_0, )$ also satisfies the property.
- Not all pairs of Banach lattices satisfy the property.

## Abstract

We prove that the class of positive operators from $L_\infty (\mu)$ to $L_1 (\nu)$ has the Bishop-Phelps-Bollob\'as property for any positive measures $\mu$ and $\nu$. The same result also holds for the pair $(c_0, \ell_1)$. We also provide an example showing that not every pair of Banach lattices satisfies the Bishop-Phelps-Bollob\'as property for positive operators.