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

Sun Kwang Kim; Han Ju Lee; Miguel Martín

doi:10.4064/sm8311-4-2016

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

Sun Kwang Kim
, Han Ju Lee
, Miguel Martín

Department of Mathematics Education

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

We study the Bishop-Phelps-Bollobás property for numerical radius (for short, BPBp-nu) of operators on ℓ₁-sums and ℓ_∞-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 X ⊕_∞ Y 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].

Original language	English
Pages (from-to)	141-151
Number of pages	11
Journal	Studia Mathematica
Volume	233
Issue number	2
DOIs	https://doi.org/10.4064/sm8311-4-2016
State	Published - 2016

Keywords

Approximation
Banach space
Bishop-Phelps-Bollobás theorem
Numerical radius attaining operators

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10.4064/sm8311-4-2016

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title = "On the Bishop-Phelps-Bollob{\'a}s theorem for operators and numerical radius",

abstract = "We study the Bishop-Phelps-Bollob{\'a}s property for numerical radius (for short, BPBp-nu) of operators on ℓ1-sums and ℓ∞-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- Bollob{\'a}s property (BPBp). On the other hand, if Y is strongly lush and X ⊕∞ Y 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].",

keywords = "Approximation, Banach space, Bishop-Phelps-Bollob{\'a}s theorem, Numerical radius attaining operators",

author = "Kim, \{Sun Kwang\} and Lee, \{Han Ju\} and Miguel Mart{\'i}n",

note = "Publisher Copyright: {\textcopyright} Instytut Matematyczny PAN, 2016.",

year = "2016",

doi = "10.4064/sm8311-4-2016",

language = "English",

volume = "233",

pages = "141--151",

journal = "Studia Mathematica",

issn = "0039-3223",

publisher = "Institute of Mathematics. Polish Academy of Sciences",

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}

TY - JOUR

T1 - On the Bishop-Phelps-Bollobás theorem for operators and numerical radius

AU - Kim, Sun Kwang

AU - Lee, Han Ju

AU - Martín, Miguel

N1 - Publisher Copyright: © Instytut Matematyczny PAN, 2016.

PY - 2016

Y1 - 2016

N2 - We study the Bishop-Phelps-Bollobás property for numerical radius (for short, BPBp-nu) of operators on ℓ1-sums and ℓ∞-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 X ⊕∞ Y 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].

AB - We study the Bishop-Phelps-Bollobás property for numerical radius (for short, BPBp-nu) of operators on ℓ1-sums and ℓ∞-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 X ⊕∞ Y 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].

KW - Approximation

KW - Banach space

KW - Bishop-Phelps-Bollobás theorem

KW - Numerical radius attaining operators

UR - https://www.scopus.com/pages/publications/84973531878

U2 - 10.4064/sm8311-4-2016

DO - 10.4064/sm8311-4-2016

M3 - Article

AN - SCOPUS:84973531878

SN - 0039-3223

VL - 233

SP - 141

EP - 151

JO - Studia Mathematica

JF - Studia Mathematica

IS - 2

ER -

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

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Keywords

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