Asymptotic Isometries for Lacunary Müntz Spaces and Applications

  • Loïc Gaillard
  • Pascal LefèvreEmail author


We prove that the normalized sequence \(\big ((p\lambda _n+1)^{1/p}t^{\lambda _n})\big )_{n\in \mathbb {Z}}\) in \(L^p([0,1])\), up to some truncation, is asymptotically isometric to the canonical basis of \(\ell ^p\) if and only if it is almost isometric if and only if \((\lambda _n+1/p)_n\) (resp. \((\lambda _n)_n\)) is a super-lacunary sequence. This extends recent results of the same authors. Similar results occur in C([0, 1]). As a particular application, we get that all the (strict) s-numbers of the classical Cesàro operator on \(L^p\) are equal to \(p'\) when \(p\in (1,+\infty )\).



The second author is partially supported by the Grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front).


  1. 1.
    AlAlam, I., Gaillard, L., Habib, G., Lefèvre, P., Maalouf, F.: Essential norm of Cesàro operator on \(L^p\) and Cesàro spaces. J. Math. Anal. Appl. 467(2), 1038–1065 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    AlAlam, I.: A Müntz space having no complement in \(L_1\). Proc. Am. Math. Soc. 136(1), 193–201 (2008)CrossRefGoogle Scholar
  3. 3.
    AlAlam, I., Habib, G., Lefèvre, P., Maalouf, F.: Essential norms of Volterra and Cesàro operators on Müntz spaces. Colloq. Math. 151(no2), 157–169 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Abrahamsen, T.A., Leraand, A., Martiny, A., Nygaard, O.: Two properties of Müntz spaces. Demonstratio Mathematica 50(1), 239–244Google Scholar
  5. 5.
    Chalendar, I., Fricain, E., Timotin, D.: Embeddings theorems for Müntz spaces. Ann. Inst. Fourier (Grenoble) 61(6), 2291–2311 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Curbera, G., Ricker, W.: Abstract Cesàro spaces: integral representations. J. Math. Anal. Appl. 441, 25–44 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dor, L.E.: On projections in \(L^1\). Ann. Math. 102(2), 463–474 (1975)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dowling, P.N., Johnson, W.B., Lennard, C.J., Turett, B.: The optimality of James’s distortion theorems. Proc. Am. Math. Soc. 125, 167–174 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fricain, E., Lefèvre, P.: \(L^2-\)Müntz spaces as model spaces. Complex Anal. Oper. Theory 13(1), 127–139 (2019)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gaillard, L., Lefèvre, P.: Lacunary Müntz spaces: isomorphisms and Carleson embeddings. Ann. Inst. Fourier 68(5), 2215–2251 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gurariy, V., Lusky, W.: Geometry of Müntz Spaces and Related Questions. Springer, Berlin (2005)CrossRefGoogle Scholar
  12. 12.
    Gurariy, V., Macaev, V.I.: Lacunary power sequences in the spaces \(C\) and \(L^p\). Am. Math. Soc. Transl. Serie 2 72, 9–21 (1966)Google Scholar
  13. 13.
    Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, London (1934)zbMATHGoogle Scholar
  14. 14.
    Lefèvre, P.: Müntz spaces and special Bloch type inequalities. In: Complex Variables and Elliptic Equations. Special issue dedicated to 130th anniversary of Vladimir I. Smirnov, pp. 1–18Google Scholar
  15. 15.
    Lefèvre, P.: The Volterra operator is finitely strictly singular from \(L^1\) to \(L^\infty \). J. Approx. Theory 214, 1–8 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Newman, D.: A Müntz space having no complement. J. Approx. Theory 40, 351–354 (1984)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Noor, W., Timotin, D.: Embeddings of Müntz spaces: the Hilbertian case. Proc. Am. Math. Soc. 141(6), 2009–2023 (2013)CrossRefGoogle Scholar
  18. 18.
    Pietsch Pietsch, A.: Operator Ideals, Vol. 20. North-Holland Mathematical Library (1980)Google Scholar
  19. 19.
    Schechtman, G.: Almost isometric \(L^p\) subspaces of \(L^p(0,1)\). J. Lond. Math. Soc. (2) 20(3), 516–528 (1979)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Laboratoire de Mathématiques de Lens (LML), EA 2462, Fédération CNRS Nord-Pas-de-Calais FR 2956Université d’ArtoisLens CedexFrance

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