On the Bishop–Phelps–Bollobás theorem for operators and numerical radius
Tom 233 / 2016
Studia Mathematica 233 (2016), 141-151
MSC: Primary 46B20; Secondary 46B04, 46B22.
DOI: 10.4064/sm8311-4-2016
Opublikowany online: 20 May 2016
Streszczenie
We study the Bishop–Phelps–Bollobás property for numerical radius (for short, BPBp-) of operators on \ell _1-sums and \ell _\infty -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\oplus _1 Y has the weak BPBp-\textrm {nu}, then (X, Y) has the Bishop–Phelps–Bollobás property (BPBp). On the other hand, if Y is strongly lush and X\oplus _\infty Y has the weak BPBp-\textrm {nu}, then (X,Y) has the BPBp. Examples of strongly lush spaces are C(K) spaces, L_1(\mu ) spaces, and finite-codimensional subspaces of C[0,1].
