On the pointwise Bishop--Phelps--Bollobás property for operators

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-Bollobás 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.

19 pages, to appear in the Canadian J. Math. In this version, section 6 and the appendix of the previous version have been removed

References in corpus (1)

• The Bishop-Phelps-Bollobás and approximate hyperplane series properties

Cited by in corpus (3)

• There is no operatorwise version of the Bishop-Phelps-Bollobás property
• The Bishop-Phelps-Bollobás property and absolute sums
• Smooth points in operator spaces and some Bishop-Phelps-Bollob$\acute{a}$s type theorems in Banach spaces