On the Pointwise Bishop–Phelps–Bollobás Property for Operators
| Autor: | Sun Kwang Kim, Vladimir Kadets, Miguel Martín, Han Ju Lee, Sheldon Dantas |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Pointwise
Pure mathematics Property (philosophy) General Mathematics 010102 general mathematics Banach space Regular polygon 46B04 (Primary) 46B07 46B20 (Secondary) Space (mathematics) Compact operator 01 natural sciences Mathematics - Functional Analysis Range (mathematics) Dimension (vector space) 0103 physical sciences 010307 mathematical physics 0101 mathematics Mathematics |
| Zdroj: | Canadian Journal of Mathematics. 71:1421-1443 |
| ISSN: | 1496-4279 0008-414X |
| DOI: | 10.4153/s0008414x18000032 |
| Popis: | 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. Comment: 19 pages, to appear in the Canadian J. Math. In this version, section 6 and the appendix of the previous version have been removed |
| Databáze: | OpenAIRE |
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