On the Pointwise Bishop–Phelps–Bollobás Property for Operators

Dantas, Sheldon  ; Kadets, Vladimir ; Kim, Sun Kwang ; Lee, Han Ju ; Martín, Miguel

Canadian journal of mathematics, Tome 71 (2019) no. 6, pp. 1421-1443

Résumé

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.

Détail

DOI : 10.4153/S0008414X18000032

Mots-clés : Banach space, norm-attaining operator, Bishop–Phelps–Bollobás property

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