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## Studia Mathematica

1996 | 118 | 2 | 101-115
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
101-115
Opis fizyczny
Daty
wydano
1996
otrzymano
1993-07-27
poprawiono
1995-07-14
Twórcy
autor
• Mathematisches Seminar der Universität Kiel, Ludewig-Meyn-Str. 4, W-2300 Kiel, 1, Germany
Bibliografia
• [COB] F. Cobos, On the Lorentz-Marcinkiewicz operator ideal, Math. Nachr. 126 (1986), 281-300.
• [DJ] M. Defant and M. Junge, On absolutely summing operators with application to the (p,q)-summing norm with few vectors, J. Funct. Anal. 103 (1992), 62-73.
• [DJ1] M. Defant and M. Junge, Random variables in weak type p spaces, Arch. Math. (Basel) 58 (1992), 399-406.
• [LET] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer, 1991.
• [LIP] W. Linde and A. Pietsch, Mappings of gaussian cylindrical measures in Banach spaces, Theory Probab. Appl. 19 (1974), 445-460.
• [LTI] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Springer, 1977.
• [LTII] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II. Function Spaces, Springer, 1979.
• [MAS] V. Mascioni, On weak cotype and weak type in Banach spaces, Note Mat. 8 (1) (1988), 67-110.
• [MAU] B. Maurey, Type et cotype dans les espaces munis d'une structure localement inconditionnelle, Sém. Maurey-Schwartz 73-74, École Polytechnique, exp. no. 24-25.
• [MSM] S. J. Montgomery-Smith, The Gaussian cotype of operators from C(K), Israel J. Math. 68 (1989), 123-128.
• [PIE] A. Pietsch, Eigenvalues and s-Numbers of Operators, Cambridge Univ. Press, 1987.
• [TAL] M. Talagrand, Cotype of operators from C(K), Invent. Math. 107 (1992), 1-40.
• [TA1] M. Talagrand, Regularity of Gaussian processes, Acta Math. 159 (1987), 99-149.
• [TJM] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Longman, 1988.
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Bibliografia
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