The Bishop-Phelps-Bollob\'as property for operators defined on $c_0$-sum of Euclidean spaces

TL;DR

This paper investigates the Bishop-Phelps-Bollobás property for operators on the $c_0$-sum of Euclidean spaces, establishing conditions under which this property holds when the target space is uniformly convex.

Contribution

It demonstrates that the pair involving the $c_0$-sum of Euclidean spaces and a uniformly convex Banach space has the BPBp for operators, extending understanding of this property.

Findings

The pair $(c_0(igoplus^{ olinebreak ext{infinity}}_{k=1} olinebreak ext{ell}_2^k),Y)$ has BPBp for operators.

The property holds when $Y$ is a uniformly convex Banach space.

Provides new insights into the structure of operator spaces with the BPBp.

Abstract

The main purpose of this paper is to study the Bishop-Phelps-Bollob\'as property for operators on $c_0$-sum of euclidean spaces. We show that the pair $ (c_0\left(\bigoplus^{\infty}_{k=1}\ell^{k}_{2} \right),Y)$ has the Bishop-Phelps-Bollob\'as property for operators (shortly BPBp for operators) whenever $Y$ is a uniformly convex Banach space.