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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

Numerical Radius

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.

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