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Ordinal indices of Small Subspaces of L p

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Abstract

We calculate the ordinal L p index defined in [3] for Rosenthal’s space X p , \({\ell_p}\) and \({\ell_2}\). We show that an infinite-dimensional subspace of L p \({(2 < p < \infty)}\) non-isomorphic to \({\ell_2}\) embeds in \({\ell_p}\) if and only if its ordinal index is the minimal possible. We also give a sufficient condition for a \({\mathcal{L}_p}\) subspace of \({\ell_p \oplus \ell_2}\) to be isomorphic to X p .

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References

  1. Alspach D.E.: Tensor products and independent sums of \({\mathcal{L}_p}\) spaces, \({1< p< \infty}\). Mem. Am. Math Soc. 138(660), viii–77 (1999)

    MathSciNet  Google Scholar 

  2. Alspach D.E.: On the complemented subspaces of X p . Israel J. Math. 74, 33–45 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bourgain J., Rosenthal H.P., Schechtman G.: An ordinal L p index for Banach spaces with an application to complemented subspaces of L p . Ann. Math. 114, 193–228 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  4. Haydon, R., Odell, E., Schulumprechet, T.: Small subspaces of L p . Ann. Math. 173, 169–209 (2011)

  5. Johnson W.B., Odell E.: Subspaces of L p which embed into \({\ell_p}\). Compos. Math. 28, 37–49 (1974)

    MathSciNet  MATH  Google Scholar 

  6. Johnson W.B., Odell E.: Subspaces and quotients of \({l_p \oplus \ell_2}\) and X p . Acta Math. 147(1-2), 117–147 (1981)

    Article  MathSciNet  Google Scholar 

  7. Kadec, M.I., Pelczynski, A.: Bases lacunary sequence and complemented subspaces in the spaces L p , Stud. Math. 21,161–176 (1961/1962)

  8. Rosenthal H.P.: On subspaces of L p (p > 2) spanned by sequensec of independent random variables. Israel J. Math. 8, 273–303 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  9. Schechtman G.: Examples of \({\mathcal{L}_p}\) spaces (\({1< p \neq 2< \infty}\)). Israel J. Math 22, 138–147 (1975)

    Article  MathSciNet  Google Scholar 

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  1. Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India

    S. Dutta & Divya Khurana

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  1. S. Dutta
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  2. Divya Khurana
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Correspondence to Divya Khurana.

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Dutta, S., Khurana, D. Ordinal indices of Small Subspaces of L p . Mediterr. J. Math. 13, 1117–1125 (2016). https://doi.org/10.1007/s00009-015-0534-2

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  • DOI: https://doi.org/10.1007/s00009-015-0534-2

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