Operators satisfying some forms of Bishop-Phelps-Bollobas type properties for norm and numerical radius

TL;DR

This paper introduces and analyzes weaker and uniform variants of Bishop-Phelps-Bollobas properties for operators and numerical radius, characterizing reflexive and weakly uniformly convex Banach spaces.

Contribution

It defines the weak and uniform Bishop-Phelps-Bollobas properties for operators and numerical radius, providing characterizations of certain Banach space classes.

Findings

Reflexive and weakly uniformly convex spaces characterized by weak BPB properties.

Identification of operator classes satisfying weak L_{o,o} property.

Extension of properties to numerical radius of bounded linear maps.

Abstract

In this paper we study a weaker form of the property $\text{\textbf{L}}_{o,o}$ called the weak $\text{\textbf{L}}_{o,o}$ and its uniform version called the weak $\text{BPB}_{\text{op}}$ which is again a weaker form the property $\text{BPB}_{\text{op}}$ for a pair of Banach spaces. We prove that a Banach space $X$ is reflexive and weakly uniformly convex if and only if the pair $(X,\mathbb{R})$ has the property weak $\text{BPB}_{\text{op}}$. We further investigate the class of all bounded linear operators from a Banach space to another Banach space satisfying the property weak $\text{\textbf{L}}_{o,o}$. Finally we introduce and study similar properties for numerical radius of a bounded linear map.