Abstract
In this article, we investigate further on the Lipschitz numerical radius and index which were recently introduced. First, we provide some renorming results on Lipschitz numerical index and introduce a concept of Lipschitz numerical radius attaining maps. Namely, we observe that for any Banach space X, the set of Lipschitz numerical indices of Banach spaces which are isomorphic to X is an interval. Moreover, we show the set of Lipschitz numerical radius attaining maps is not dense in the space of Lipschitz maps vanishing at zero. Next, we discuss the Lipschitz numerical index of vector-valued function spaces, absolute sums of Banach spaces, the Köthe–Bochner spaces, and Banach spaces which contain a dense union of increasing family of one-complemented subspaces.
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Acknowledgements
The authors are thankful to the anonymous referee for several helpful suggestions.
Funding
G. Choi was supported by a Research promotion program of SCNU. M. Jung 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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