On the Bishop–Phelps–Bollobás theorem for operators and numerical radius

Sun Kwang Kim¹ ; Han Ju Lee² ; Miguel Martín³

¹ Department of Mathematics Kyonggi University Suwon 443-760, Republic of Korea
² Department of Mathematics Education Dongguk University – Seoul 04620 Seoul, Republic of Korea
³ Departamento de Análisis Matemático Facultad de Ciencias Universidad de Granada E-18071 Granada, Spain

Studia Mathematica, Tome 233 (2016) no. 2, pp. 141-151

Résumé

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 $C[0,1]$.

Détail

DOI : 10.4064/sm8311-4-2016

Keywords: study bishop phelps bollob property numerical radius short bpbp textrm operators ell sums ell infty sums banach spaces precisely introduce property banach spaces which call strongly lush strongly lush oplus has weak bpbp textrm has bishop phelps bollob property bpbp other strongly lush oplus infty has weak bpbp textrm has bpbp examples strongly lush spaces spaces spaces finite codimensional subspaces