We study the Bishop-Phelps-Bollobás property for numerical radius (for short, BPBp-nu) of operators on ℓ₁-sums and -sums of Banach spaces. More precisely, we introduce a property of Banach spaces, which we call strongly lush. We find that if X is strongly lush and X ⊕₁ Y has the weak BPBp-nu, then (X,Y) has the Bishop-Phelps-Bollobás property (BPBp). On the other hand, if Y is strongly lush and has the weak BPBp-nu, then (X,Y) has the BPBp. Examples of strongly lush spaces are C(K) spaces, L₁(μ) spaces, and finite-codimensional subspaces of C[0,1].
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm8311-4-2016,
author = {Sun Kwang Kim and Han Ju Lee and Miguel Mart\'\i n},
title = {On the Bishop-Phelps-Bollob\'as theorem for operators and numerical radius},
journal = {Studia Mathematica},
volume = {233},
year = {2016},
pages = {141-151},
zbl = {06586872},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8311-4-2016}
}
Sun Kwang Kim; Han Ju Lee; Miguel Martín. On the Bishop-Phelps-Bollobás theorem for operators and numerical radius. Studia Mathematica, Tome 233 (2016) pp. 141-151. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8311-4-2016/