𝐿_{𝑝}+𝐿_{∞} and 𝐿_{𝑝}∩𝐿_{∞} are not isomorphic for all 1≤𝑝<∞, 𝑝≠2

Astashkin, Sergey; Maligranda, Lech

doi:10.1090/proc/13928

$L_p +L_{\infty }$ and $L_p \cap L_{\infty }$ are not isomorphic for all $1 \leq p < \infty$, $p \neq 2$
HTML articles powered by AMS MathViewer

by Sergey V. Astashkin and Lech Maligranda PDF

Proc. Amer. Math. Soc. 146 (2018), 2181-2194 Request permission

Abstract:

We prove the result stated in the title. It comes as a consequence of the fact that the space $L_p \cap L_{\infty }$, $1\leq p<\infty$, $p\neq 2$, does not contain a complemented subspace isomorphic to $L_p$. In particular, as a subproduct, we show that $L_p \cap L_{\infty }$ contains a complemented subspace isomorphic to ${\ell }_2$ if and only if $p = 2$.

References

Fernando Albiac and Nigel J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR 2192298
S. V. Astashkin, K. Leśnik, and L. Maligranda, Isomorphic structure of Cesàro and Tandori spaces, Canad. J. Math. (to appear). Preprint of 33 pages submitted on 10 December 2015 at arXiv:1512.03336
Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275, DOI 10.1007/978-3-642-66451-9
S. J. Dilworth, Intersection of Lebesgue spaces $L_1$ and $L_2$, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1185–1188. MR 955005, DOI 10.1090/S0002-9939-1988-0955005-3
S. J. Dilworth, A scale of linear spaces related to the $L_p$ scale, Illinois J. Math. 34 (1990), no. 1, 140–158. MR 1031891, DOI 10.1215/ijm/1255988499
A. Grothendieck, Sur certains sous-espaces vectoriels de $L^p$, Canad. J. Math. 6 (1954), 158–160 (French). MR 58867, DOI 10.4153/cjm-1954-017-x
James Hagler and Charles Stegall, Banach spaces whose duals contain complemented subspaces isomorphic to $C[0,1]$, J. Functional Analysis 13 (1973), 233–251. MR 0350381, DOI 10.1016/0022-1236(73)90033-5
W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 19 (1979), no. 217, v+298. MR 527010, DOI 10.1090/memo/0217
N. J. Kalton, Lattice structures on Banach spaces, Mem. Amer. Math. Soc. 103 (1993), no. 493, vi+92. MR 1145663, DOI 10.1090/memo/0493
S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs. MR 649411
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367, DOI 10.1007/978-3-662-35347-9
Lech Maligranda, The $K$-functional for symmetric spaces, Interpolation spaces and allied topics in analysis (Lund, 1983) Lecture Notes in Math., vol. 1070, Springer, Berlin, 1984, pp. 169–182. MR 760482, DOI 10.1007/BFb0099100
Lech Maligranda, The $K$-functional for $p$-convexifications, Positivity 17 (2013), no. 3, 707–710. MR 3090688, DOI 10.1007/s11117-012-0200-x
Yves Raynaud, Complemented Hilbertian subspaces in rearrangement invariant function spaces, Illinois J. Math. 39 (1995), no. 2, 212–250. MR 1316534
Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
C. Stegall, Banach spaces whose duals contain $l_{1}(\Gamma )$ with applications to the study of dual $L_{1}(\mu )$ spaces, Trans. Amer. Math. Soc. 176 (1973), 463–477. MR 315404, DOI 10.1090/S0002-9947-1973-0315404-3
S. J. Szarek, On the best constants in the Khinchin inequality, Studia Math. 58 (1976), no. 2, 197–208. MR 430667, DOI 10.4064/sm-58-2-197-208