Comparing gaussian and Rademacher cotype for operators on the space of continuous functions

Marius Junge

Studia Mathematica (1996)

  • Volume: 118, Issue: 2, page 101-115
  • ISSN: 0039-3223

Abstract

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We prove an abstract comparison principle which translates gaussian cotype into Rademacher cotype conditions and vice versa. More precisely, let 2 < q < ∞ and T: C(K) → F a continuous linear operator. (1) T is of gaussian cotype q if and only if ( ∑ k ( ( ∥ T x k ∥ F ) / ( √ l o g ( k + 1 ) ) ) q ) 1 / q ≤ c ∥ ∑ k ɛ k x k ∥ L 2 ( C ( K ) ) , for all sequences ( x k ) k ∈ ℕ ⊂ C ( K ) with ( ∥ T x k ∥ ) k = 1 n decreasing. (2) T is of Rademacher cotype q if and only if ( ∑ k ( ∥ T x k ∥ F √ ( ( l o g ( k + 1 ) ) q ) ) 1 / q ≤ c ∥ ∑ k g k x k ∥ L 2 ( C ( K ) ) , for all sequences ( x k ) k ∈ ℕ ⊂ C ( K ) with ( ∥ T x k ∥ ) k = 1 n decreasing. Our method allows a restriction to a fixed number of vectors and complements the corresponding results of Talagrand.

How to cite

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Junge, Marius. "Comparing gaussian and Rademacher cotype for operators on the space of continuous functions." Studia Mathematica 118.2 (1996): 101-115. <http://eudml.org/doc/216266>.

@article{Junge1996,
abstract = {We prove an abstract comparison principle which translates gaussian cotype into Rademacher cotype conditions and vice versa. More precisely, let 2 < q < ∞ and T: C(K) → F a continuous linear operator. (1) T is of gaussian cotype q if and only if $(∑_k ((∥Tx_k∥_F)/(√log(k+1)))^q)^\{1/q\} ≤ c ∥ ∑_k ɛ_\{k\} x_\{k\} ∥_\{L_\{2\}(C(K))\}$, for all sequences $(x_k)_\{k∈ℕ\} ⊂ C(K)$ with $(∥Tx_k∥)_\{k=1\}^n$ decreasing. (2) T is of Rademacher cotype q if and only if $(∑_k (∥Tx_k∥_\{F\} √((log(k+1))^q) )^\{1/q\} ≤ c ∥∑_k g_\{k\}x_\{k\}∥_\{L_2(C(K))\}$, for all sequences $(x_k)_\{k∈ℕ\} ⊂ C(K)$ with $(∥Tx_k∥)_\{k=1\}^n$ decreasing. Our method allows a restriction to a fixed number of vectors and complements the corresponding results of Talagrand.},
author = {Junge, Marius},
journal = {Studia Mathematica},
keywords = {abstract comparison principle; Gaussian cotype; Rademacher cotype},
language = {eng},
number = {2},
pages = {101-115},
title = {Comparing gaussian and Rademacher cotype for operators on the space of continuous functions},
url = {http://eudml.org/doc/216266},
volume = {118},
year = {1996},
}

TY - JOUR
AU - Junge, Marius
TI - Comparing gaussian and Rademacher cotype for operators on the space of continuous functions
JO - Studia Mathematica
PY - 1996
VL - 118
IS - 2
SP - 101
EP - 115
AB - We prove an abstract comparison principle which translates gaussian cotype into Rademacher cotype conditions and vice versa. More precisely, let 2 < q < ∞ and T: C(K) → F a continuous linear operator. (1) T is of gaussian cotype q if and only if $(∑_k ((∥Tx_k∥_F)/(√log(k+1)))^q)^{1/q} ≤ c ∥ ∑_k ɛ_{k} x_{k} ∥_{L_{2}(C(K))}$, for all sequences $(x_k)_{k∈ℕ} ⊂ C(K)$ with $(∥Tx_k∥)_{k=1}^n$ decreasing. (2) T is of Rademacher cotype q if and only if $(∑_k (∥Tx_k∥_{F} √((log(k+1))^q) )^{1/q} ≤ c ∥∑_k g_{k}x_{k}∥_{L_2(C(K))}$, for all sequences $(x_k)_{k∈ℕ} ⊂ C(K)$ with $(∥Tx_k∥)_{k=1}^n$ decreasing. Our method allows a restriction to a fixed number of vectors and complements the corresponding results of Talagrand.
LA - eng
KW - abstract comparison principle; Gaussian cotype; Rademacher cotype
UR - http://eudml.org/doc/216266
ER -

References

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  10. [MSM] S. J. Montgomery-Smith, The Gaussian cotype of operators from C(K), Israel J. Math. 68 (1989), 123-128. Zbl0701.47006
  11. [PIE] A. Pietsch, Eigenvalues and s-Numbers of Operators, Cambridge Univ. Press, 1987. 
  12. [TAL] M. Talagrand, Cotype of operators from C(K), Invent. Math. 107 (1992), 1-40. Zbl0788.47022
  13. [TA1] M. Talagrand, Regularity of Gaussian processes, Acta Math. 159 (1987), 99-149. Zbl0712.60044
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