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

Tom 233 / 2016

Sun Kwang Kim, Han Ju Lee, Miguel Martín 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 $C[0,1]$.

Autorzy

  • Sun Kwang KimDepartment of Mathematics
    Kyonggi University
    Suwon 443-760, Republic of Korea
    e-mail
  • Han Ju LeeDepartment of Mathematics Education
    Dongguk University – Seoul
    04620 Seoul, Republic of Korea
    e-mail
  • Miguel MartínDepartamento de Análisis Matemático
    Facultad de Ciencias
    Universidad de Granada
    E-18071 Granada, Spain
    e-mail

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