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Title

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

Authors 1

Affiliations

  1. Mathematisches Seminar der Universität Kiel, Ludewig-Meyn-Str. 4, W-2300 Kiel, 1, Germany

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(∥Txk∥F√log(k+1))q)1q≤c∥∑kɛkxk∥L2(C(K)), for all sequences (xk)k∈ℕ⊂C(K) with (∥Txk∥)k=1n decreasing. (2) T is of Rademacher cotype q if and only if (∑k(∥Txk∥F√((log(k+1))q))1q≤c∥∑kgkxk∥L2(C(K)), for all sequences (xk)k∈ℕ⊂C(K) with (∥Txk∥)k=1n decreasing. Our method allows a restriction to a fixed number of vectors and complements the corresponding results of Talagrand.

Bibliography

  1. [COB] F. Cobos, On the Lorentz-Marcinkiewicz operator ideal, Math. Nachr. 126 (1986), 281-300.
  2. [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.
  3. [DJ1] M. Defant and M. Junge, Random variables in weak type p spaces, Arch. Math. (Basel) 58 (1992), 399-406.
  4. [LET] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer, 1991.
  5. [LIP] W. Linde and A. Pietsch, Mappings of gaussian cylindrical measures in Banach spaces, Theory Probab. Appl. 19 (1974), 445-460.
  6. [LTI] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Springer, 1977.
  7. [LTII] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II. Function Spaces, Springer, 1979.
  8. [MAS] V. Mascioni, On weak cotype and weak type in Banach spaces, Note Mat. 8 (1) (1988), 67-110.
  9. [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.
  10. [MSM] S. J. Montgomery-Smith, The Gaussian cotype of operators from C(K), Israel J. Math. 68 (1989), 123-128.
  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.
  13. [TA1] M. Talagrand, Regularity of Gaussian processes, Acta Math. 159 (1987), 99-149.
  14. [TJM] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Longman, 1988.
Studia Mathematica
1996/118/2
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Pages:
101-115
Main language of publication
English
Received
1993-07-27
Accepted
1995-07-14
Published
1996
Exact and natural sciences