Abstract
We show that if a bounded linear operator can be approximated by a net (or sequence) of uniformly bounded finite rank Lipschitz mappings pointwisely, then it can be approximated by a net (or sequence) of uniformly bounded finite rank linear operators under the strong operator topology. As an application, we deduce that a Banach space has an (unconditional) Lipschitz frame if and only if it has an (unconditional) Schauder frame. Another immediate consequence of our result recovers the famous Godefroy-Kalton theorem (Godefroy and Kalton (2003)) which says that the Lipschitz bounded approximation property and the bounded approximation property are equivalent for every Banach space.
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Acknowledgements
Rui Liu and Jie Shen were supported by National Natural Science Foundation of China (Grant Nos. 11671214, 11971348 and 12071230), Hundred Young Academia Leaders Program of Nankai University (Grant Nos. 63223027 and ZB22000105), Undergraduate Education and Teaching Project of Nankai University (Grant No. NKJG2022053) and National College Students’ Innovation and Entrepreneurship Training Program of Nankai University (Grant No. 202210055048). Bentuo Zheng was supported by Simons Foundation (Grant No. 585081). The authors express their great appreciation to the referees for helpful suggestions which led to the current version of this paper.
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Liu, R., Shen, J. & Zheng, B. Operators with the Lipschitz bounded approximation property. Sci. China Math. 66, 1545–1554 (2023). https://doi.org/10.1007/s11425-022-2000-y
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DOI: https://doi.org/10.1007/s11425-022-2000-y