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Some remarks about disjointly homogeneous symmetric spaces

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Abstract

Let \(1\le p<\infty \). A symmetric space X on [0, 1] is said to be p-disjointly homogeneous (resp. restricted p-disjointly homogeneous) if every sequence of normalized pairwise disjoint functions from X (resp. characteristic functions) contains a subsequence equivalent in X to the unit vector basis of \(l_p\). Answering a question posed in the paper (Hernández et al. in Funct Approx 50(2):215–232, 2014), we construct, for each \(1\le p<\infty \), a restricted p-disjointly homogeneous symmetric space, which is not p-disjointly homogeneous. Moreover, we prove that the property of p-disjoint homogeneity is preserved under Banach isomorphisms.

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Notes

  1. Recently, this problem was solved in the negative (see [5])

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Acknowledgements

The author would like to thank the referees for their extremely valuable and very helpful remarks and comments.

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  1. Department of Mathematics, Samara National Research University, Moskovskoye shosse 34, Samara, Russia, 443086

    Sergey V. Astashkin

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  1. Sergey V. Astashkin
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Correspondence to Sergey V. Astashkin.

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The work was supported by the Ministry of Education and Science of the Russian Federation, Project 1.470.2016/1.4 and by the RFBR Grant 18-01-00414.

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Astashkin, S.V. Some remarks about disjointly homogeneous symmetric spaces. Rev Mat Complut 32, 823–835 (2019). https://doi.org/10.1007/s13163-018-0289-y

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