The Bishop-Phelps-Bollob{\'a}s property for numerical radius of operators on $L_1 (\mu)$

TL;DR

This paper introduces the Bishop-Phelps-Bollobás property for numerical radius in operator spaces on L1 spaces and demonstrates its presence in various subspaces, including finite-rank, compact, and weakly compact operators.

Contribution

It defines the BPBp-ν for operators on L1 spaces and proves its validity for several important subspaces, expanding understanding of operator properties.

Findings

Subspaces of L(L1(μ)) have BPBp-ν for finite measures.

Finite-rank, compact, and weakly compact operators on L1(μ) possess BPBp-ν.

The paper extends the theory of Bishop-Phelps-Bollobás properties to numerical radius.

Abstract

In this paper, we introduce the notion of the Bishop-Phelps-Bollob\'as property for numerical radius (BPBp-$\nu$) for a subclass of the space of bounded linear operators. Then, we show that certain subspaces of $\mathcal{L}(L_1(\mu))$ have the BPBp-$\nu$ for every finite measure $\mu $. As a consequence we deduce that the subspaces of finite-rank operators, compact operators and weakly compact operators on $L_1(\mu)$ have the BPBp-$\nu$.