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-$\textrm {nu}$) 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 $C[0,1]$.
