Abstract
Necessary and sufficient conditions are found under which a symmetric space X on [0,1] of type 2 has the following property, which was first proved for the spaces Lp, p > 2, by Kadets and Pełczyński: if \(\left\{ {{u_n}} \right\}_{n = 1}^\infty \) is an unconditional basic sequence in X such that
then the norms of the spaces X and L1 are equivalent on the closed linear span [un] in X. For sequences of martingale differences, this implication holds in any symmetric space of type 2.
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Funding
This work was prepared in the framework of the implementation of the state task of the Ministry of Education and Science of the Russian Federation (project no. 1.470.2016/1.4) and also supported in part by the Russian Foundation for Basic Research (grant no. 18-01-00414-a).
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Russian Text © The Author(s), 2019, published in Matematicheskie Zametki, 2019, Vol. 106, No. 2, pp. 174–187.
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Astashkin, S.V. On a Theorem of Kadets and Pełczyński. Math Notes 106, 172–182 (2019). https://doi.org/10.1134/S0001434619070216
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DOI: https://doi.org/10.1134/S0001434619070216