On various types of density of numerical radius attaining operators

TL;DR

This paper investigates the denseness of numerical radius attaining operators in Banach spaces, introducing properties like BPBpp-nu and BPBop-nu, and explores their implications and restrictions across different space types.

Contribution

It establishes conditions under which Banach spaces satisfy BPBpp-nu and BPBop-nu, and relates these properties to space dimensionality, numerical index, and subdifferentiability.

Findings

Spaces with micro-transitive norm and positive second numerical index satisfy BPBpp-nu.

BPBop-nu is highly restrictive, holding only in one-dimensional spaces under general assumptions.

Finite-dimensional spaces and c0 satisfy local BPBpp-nu; positive numerical index and approximation property imply finite dimensionality.

Abstract

In this paper, we are interested in studying two properties related to the denseness of the operators which attain their numerical radius: the Bishop-Phelps-Bollob\'as point and operator properties for numerical radius (BPBpp-nu and BPBop-nu, respectively). We prove that every Banach space with micro-transitive norm and second numerical index strictly positive satisfy the BPBpp-nu and that, if the numerical index of $X$ is 1, only one-dimensional spaces enjoy it. On the other hand, we show that the BPBop-nu is a very restrictive property: under some general assumptions, it holds only for one-dimensional spaces. We also consider two weaker properties, the local versions of BPBpp-nu and BPBop-nu, where the $\eta$ which appears in their definition does not depend just on $\epsilon > 0$ but also on a state $(x, x^*)$ or on a numerical radius one operator $T$. We address the relation between the local BPBpp-nu and the strong subdifferentiability of the norm of the space $X$. We show that finite dimensional spaces and $c_0$ are examples of Banach spaces satisfying the local BPBpp-nu, and we exhibit an example of a Banach space with strongly subdifferentiable norm failing it. We finish the paper by showing that finite dimensional spaces satisfy the local BPBop-nu and that, if $X$ has strictly positive numerical index and has the approximation property, this property is equivalent to finite dimensionality.