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A generalized ACK structure and the denseness of norm attaining operators

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Inspired by the recent work of Cascales et al., we introduce a generalized concept of ACK structure on Banach spaces. Using this property, which we call by the quasi-ACK structure, we are able to extend known universal properties on range spaces concerning the density of norm attaining operators. We provide sufficient conditions for quasi-ACK structure of spaces and results on the stability of quasi-ACK structure. As a consequence, we present new examples satisfying (Lindenstrauss) property B\(^k\), which have not been known previously. We also prove that property B\(^k\) is stable under injective tensor products in certain cases. Moreover, ACK structure of some Banach spaces of vector-valued holomorphic functions is also discussed, leading to new examples of universal BPB range spaces for certain operator ideals.

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Acknowledgements

The authors would like to thank Sun Kwang Kim and Miguel Martín for valuable comments and remarks leading to improvement of this paper. The authors also want to thank anonymous referees for their careful reading and helpful suggestions. The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2022R1A6A3A01086079) and by the Ministry of Education, Science and Technology [NRF-2020R1A2C1A01010377]. The second author was supported by NRF [NRF-2019R1A2C1003857], by POSTECH Basic Science Research Institute Grant [NRF-2021R1A6A1A10042944] and by a KIAS Individual Grant [MG086601] at Korea Institute for Advanced Study.

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  1. Department of Mathematics Education, Dongguk University, Seoul, 04620, Republic of Korea

    Geunsu Choi

  2. School of Mathematics, Korea Institute for Advanced Study, Seoul, 02455, Republic of Korea

    Mingu Jung

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  1. Geunsu Choi
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Correspondence to Mingu Jung.

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Choi, G., Jung, M. A generalized ACK structure and the denseness of norm attaining operators. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 87 (2023). https://doi.org/10.1007/s13398-023-01421-x

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