TY - JOUR
T1 - On the pointwise Bishop-Phelps-Bollobás property for operators
AU - Dantas, Sheldon
AU - Kadets, Vladimir
AU - Kim, Sun Kwang
AU - Lee, Han Ju
AU - Martín, Miguel
N1 - Publisher Copyright:
© Canadian Mathematical Society 2018
PY - 2019
Y1 - 2019
N2 - 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 Lp(µ) 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.
AB - 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 Lp(µ) 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.
KW - Banach space
KW - Bishop-Phelps-Bollobás property
KW - Norm-attaining operator
UR - http://www.scopus.com/inward/record.url?scp=85059301782&partnerID=8YFLogxK
U2 - 10.4153/S0008414X18000032
DO - 10.4153/S0008414X18000032
M3 - Article
AN - SCOPUS:85059301782
SN - 0008-414X
VL - 71
SP - 1421
EP - 1443
JO - Canadian Journal of Mathematics
JF - Canadian Journal of Mathematics
IS - 6
ER -