The Rademacher cotype of operators from 𝑙^{𝑁}_{∞}

Montgomery-Smith, S. J.; Talagrand, M.

doi:10.1090/S0002-9939-1991-1043416-X

The Rademacher cotype of operators from $l\sp N\sb \infty$

Authors: S. J. Montgomery-Smith and M. Talagrand
Journal: Proc. Amer. Math. Soc. 112 (1991), 187-194
MSC: Primary 46B20; Secondary 47B10, 47B37, 60G99
DOI: https://doi.org/10.1090/S0002-9939-1991-1043416-X
MathSciNet review: 1043416
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Abstract: We show that for any operator $T:l_\infty ^N \to Y$ , where is a Banach space, its cotype 2 constant, ${K^{(2)}}(T)$ , is related to its -summing norm, ${\pi _{2,1}}(T)$ , by

$\displaystyle {K^{(2)}}(T) \leq c\operatorname{log} \operatorname{log} N{\pi _{2,1}}(T).$

Thus, we can show that there is an operator $T:C(K) \to Y$ that has cotype 2, but is not

-summing.

[C] J. Creekmore, Type and cotype in Lorentz 𝐿_{𝑝𝑞} spaces, Nederl. Akad. Wetensch. Indag. Math. 43 (1981), no. 2, 145–152. MR 707247
[H] Richard A. Hunt, On 𝐿(𝑝,𝑞) spaces, Enseignement Math. (2) 12 (1966), 249–276. MR 0223874
[J] G. J. O. Jameson, Relations between summing norms of mappings of 𝑙ⁿ_{∞}, Math. Z. 194 (1987), no. 1, 89–94. MR 871220, https://doi.org/10.1007/BF01168007
[K] S. Kwapień, On a theorem of L. Schwartz and its applications to absolutely summing operators, Studia Math. 38 (1970), 193–201. (errata insert). MR 0278090, https://doi.org/10.4064/sm-38-1-193-201
[LdT] Michel Ledoux and Michel Talagrand, Probability in Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23, Springer-Verlag, Berlin, 1991. Isoperimetry and processes. MR 1102015
[LT1] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. MR 0500056
[LT2] 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
[Ma1] B. Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans un espace ${L^p}$ , Astérisque 11 (1974).
[Ma2] -, Type et cotype dans les espaces munis de structures locales inconditionelles, Seminaire Maurey-Schwartz, 1973-74, Exp. 24-25.
[Mo1] S. J. Montgomery-Smith, The cotype of operators from , Ph.D. thesis, Cambridge, August 1988.
[Mo2] S. J. Montgomery-Smith, The Gaussian cotype of operators from 𝐶(𝐾), Israel J. Math. 68 (1989), no. 1, 123–128. MR 1035886, https://doi.org/10.1007/BF02764974
[P1] Gilles Pisier, Factorization of linear operators and geometry of Banach spaces, CBMS Regional Conference Series in Mathematics, vol. 60, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 829919
[P2] Gilles Pisier, Factorization of operators through 𝐿_{𝑝∞} or 𝐿_{𝑝1} and noncommutative generalizations, Math. Ann. 276 (1986), no. 1, 105–136. MR 863711, https://doi.org/10.1007/BF01450929
[T] Michel Talagrand, The canonical injection from 𝐶([0,1]) into 𝐿_{2,1} is not of cotype 2, Banach space theory (Iowa City, IA, 1987) Contemp. Math., vol. 85, Amer. Math. Soc., Providence, RI, 1989, pp. 513–521. MR 983403, https://doi.org/10.1090/conm/085/983403