Geodesic
On the Pointwise Bishop–Phelps–Bollobás Property for Operators
Canadian journal of mathematics, Tome 71 (2019) no. 6, pp. 1421-1443

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DOI

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.
DOI : 10.4153/S0008414X18000032
Mots-clés : Banach space, norm-attaining operator, Bishop–Phelps–Bollobás property 
Dantas, Sheldon; Kadets, Vladimir; Kim, Sun Kwang; Lee, Han Ju; Martín, Miguel. On the Pointwise Bishop–Phelps–Bollobás Property for Operators. Canadian journal of mathematics, Tome 71 (2019) no. 6, pp. 1421-1443. doi: 10.4153/S0008414X18000032
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     title = {On the {Pointwise} {Bishop{\textendash}Phelps{\textendash}Bollob\'as} {Property} for {Operators}},
     journal = {Canadian journal of mathematics},
     pages = {1421--1443},
     year = {2019},
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