On the pointwise Bishop--Phelps--Bollob\'as property for operators

TL;DR

This paper investigates the pointwise Bishop-Phelps-Bollobás property for operators between Banach spaces, identifying conditions under which certain spaces have this property and exploring its limitations for specific classes of spaces.

Contribution

It introduces the concepts of universal pointwise BPB domain and range spaces, characterizes their properties, and analyzes their existence and limitations across various Banach spaces.

Findings

Universal pointwise BPB domain spaces are uniformly convex.

L_p spaces with p>2 do not have the property.

Spaces that are both uniformly convex and smooth fail the property if dimension > 1.

Abstract

We study approximation of operators between Banach spaces $X$ and $Y$ that nearly attain their norms in a given point by operators that attain their norms at the same point. When such approximations exist, we say that the pair $(X, Y)$ has the pointwise Bishop-Phelps-Bollob\'as property (pointwise BPB property for short). In this paper we mostly concentrate on those $X$, called universal pointwise BPB domain spaces, such that $(X, Y)$ possesses pointwise BPB property for every $Y$, and on those $Y$, called universal pointwise BPB range spaces, such that $(X, Y)$ enjoys pointwise BPB property for every uniformly smooth $X$. We show that every universal pointwise BPB domain space is uniformly convex and that $L_p(\mu)$ spaces fail to have this property when $p>2$. For universal pointwise BPB range space, we show that every simultaneously uniformly convex and uniformly smooth Banach space fails it if its dimension is greater than one. We also discuss a version of the pointwise BPB property for compact operators.