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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 $ Y$ is a Banach space, its cotype 2 constant, $ {K^{(2)}}(T)$, is related to its $ (2,1)$-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 $ 2$-summing.

References [Enhancements On Off] (What's this?)

  • [C] J. Creekmore, Type and cotype in Lorentz $ {L_{p,q}}$ spaces, Indag. Math. 43 (1981), 145-152. MR 707247 (84i:46032)
  • [H] R. A. Hunt, On $ L(p,q)$ spaces, Enseign. Math. (2) 12 (1966), 249-275. MR 0223874 (36:6921)
  • [J] G. J. O. Jameson, Relations between summing norms of mappings on $ l_\infty ^n$, Math. Z. 194 (1987), 89-94. MR 871220 (88d:47026)
  • [K] S. Kwapien, On a theorem of L. Schwartz and its applications to absolutely summing operators, Studia. Math. 38 (1984), 193-200. MR 0278090 (43:3822)
  • [LdT] M. Ledoux and M. Talagrand, Isoperimetry and processes in probability in a Banach space, Springer-Verlag (to appear). MR 1102015 (93c:60001)
  • [LT1] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I--sequence spaces, Springer-Verlag, 1977. MR 0500056 (58:17766)
  • [LT2] -, Classical Banach spaces II--function spaces, Springer-Verlag, 1979. MR 540367 (81c:46001)
  • [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 $ C(K)$, Ph.D. thesis, Cambridge, August 1988.
  • [Mo2] -, The Gaussian cotype of operators from $ C(K)$, Israel J. Math. 68 (1989), 123-128. MR 1035886 (91c:46015)
  • [P1] G. Pisier, Factorization of linear operators and geometry of banach spaces, Amer. Math. Soc., Providence, RI, 1986. MR 829919 (88a:47020)
  • [P2] -, Factorization of operators through $ {L_{p\infty }}$ or $ {L_{p1}}$ and non-commutative generalizations, Math. Ann. 276 (1986), 105-136. MR 863711 (88f:47013)
  • [T] M. Talagrand, The canonical injection from $ C([0,1])$ into $ {L_{2,1}}$ is not of cotype 2, Contemp. Math. 85 (1989), 513-521. MR 983403 (90b:46064)

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DOI: https://doi.org/10.1090/S0002-9939-1991-1043416-X
Article copyright: © Copyright 1991 American Mathematical Society

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