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On the Pointwise Bishop–Phelps–Bollobás Property for Operators
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 71 / Issue 6 / December 2019
- Published online by Cambridge University Press:
- 17 October 2018, pp. 1421-1443
- Print publication:
- December 2019
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- Article
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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 
$L_{p}(\unicode[STIX]{x1D707})$ spaces fail to have this property when 
$p>2$. No universal pointwise BPB range space can be simultaneously uniformly convex and uniformly smooth unless its dimension is one. We also discuss a version of the pointwise BPB property for compact operators.
