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On the Compact Operators Case of the Bishop–Phelps–Bollobás Property for Numerical Radius

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Abstract

We study the Bishop–Phelps–Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that $$C_0(L)$$ spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space L. To this end, on the one hand, we provide some techniques allowing to pass the BPBp-nu for compact operators from subspaces to the whole space and, on the other hand, we prove some strong approximation property of $$C_0(L)$$ spaces and their duals. Besides, we also show that real Hilbert spaces and isometric preduals of $$\ell _1$$ have the BPBp-nu for compact operators.
Results Math (2021) 76:122
c
2021 The Author(s), under exclusive licence to
Springer Nature Switzerland AG
1422-6383/21/030001-23
published online May 26, 2021
https://doi.org/10.1007/s00025-021-01430-5 Results in Mathematics
On the Compact Operators Case of the
Bishop–Phelps–Bollob´as Property for
Domingo Garc´ıa , Manuel Maestre ,MiguelMart´ın ,and
´
Oscar Rold´an
Abstract. We study the Bishop–Phelps–Bollob´as property for numerical
radius restricted to the case of compact operators (BPBp-nu for com-
pact operators in short). We show that C0(L) spaces have the BPBp-nu
for compact operators for every Hausdorﬀ topological locally compact
space L. To this end, on the one hand, we provide some techniques allow-
ing to pass the BPBp-nu for compact operators from subspaces to the
whole space and, on the other hand, we prove some strong approxima-
tion property of C0(L) spaces and their duals. Besides, we also show that
real Hilbert spaces and isometric preduals of 1have the BPBp-nu for
compact operators.
Mathematics Subject Classiﬁcation. Primary 46B04, Secondary 46B20,
46B25, 46B28.
Keywords. Banach space, compact operator, Bishop–Phelps–Bollob´as prop-
erty, numerical radius attaining operator, approximation property.
The ﬁrst and second authors were supported by MINECO and FEDER Project MTM2017-
83262-C2-1-P and by Prometeo PROMETEO/2017/102. The third author was supported
by Projects PGC2018-093794-B-I00 (MCIU/AEI/FEDER, UE), A-FQM-484-UGR18 (Uni-
versidad de Granada and Junta de Analuc´ıa/FEDER, UE), and FQM-185 (Junta de An-
daluc´ıa/FEDER, UE). The fourth author was supported by the Spanish Ministerio de Cien-
cia, Innovaci´on y Universidades, Grant FPU17/02023, and by MINECO and FEDER Project
MTM2017-83262-C2-1-P.
Chapter
In this survey, we provide an overview from 2008 to 2021 about the Bishop–Phelps–Bollobás theorem.KeywordsNorm attaining operatorsBishop–Phelps theoremBishop–Phelps–Bollobás property
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