The strong Bishop-Phelps-Bollob\'as property

TL;DR

This paper introduces the strong Bishop-Phelps-Bollobás property (sBPBp) for bounded linear operators between Banach spaces, explores which pairs satisfy it, and characterizes it for specific classical spaces.

Contribution

It defines the sBPBp for operators, provides positive and negative examples of Banach space pairs, and characterizes the property for pairs of ll_p and ll_q spaces.

Findings

Pairs of Banach spaces satisfying sBPBp are identified.

Concrete examples of pairs failing sBPBp are provided.

Complete characterization of sBPBp for (ll_p, ll_q) pairs is achieved.

Abstract

In this paper we introduce the strong Bishop-Phelps-Bollob\'as property (sBPBp) for bounded linear operators between two Banach spaces $X$ and $Y$. This property is motivated by a Kim-Lee result which states, under our notation, that a Banach space $X$ is uniformly convex if and only if the pair $(X,\mathbb{K})$ satisfies the sBPBp. Positive results of pairs of Banach spaces $(X,Y)$ satisfying this property are given and concrete pairs of Banach spaces $(X, Y)$ failing it are exhibited. A complete characterization of the sBPBp for the pairs $(\ell_p, \ell_q)$ is also provided.