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On R-boundedness of Unions of Sets of Operators

  • Onno van Gaans
Part of the Operator Theory: Advances and Applications book series (OT, volume 168)

Abstract

It is shown that the union of a sequence T 1, T 2, . . . of R-bounded sets of operators from X into Y with R-bounds T 1, T 2, . . ., respectively, is R-bounded if X is a Banach space of cotype q, Y a Banach space of type p, and Σk=1/∞ T k/r < ∞, where r = pq/(q − p) if q < ∞ and r = p if q = ∞. Here 1 ≤ p ≤ 2 ≤ q ≤ ∞ and pq. The power r is sharp. Each Banach space that contains an isomorphic copy of c 0 admits operators T 1, T 2, . . . such that ∥T k∥ = 1/k, k ∈ ℕ, and T 1, T 2, . . . is not R-bounded. Further it is shown that the set of positive linear contractions in an infinite-dimensional L p is R-bounded only if p = 2.

Keywords

Banach Space Measure Space Banach Lattice Maximal Regularity Isomorphic Copy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    W. Arendt and S. Bu, The operator-valued Marcinkiewicz multiplier theorem and maximal regularity, Math. Z. 240 (2002), no. 2, 311–343.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    W. Arendt and S. Bu, Tools for maximal regularity, Math. Proc. Cambridge Philos. Soc. 134 (2003), no. 2, 317–336.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    S. Blunck, Maximal regularity of discrete and continuous time evolution equations, Studia Math. 146 (2001), no. 2, 157–176.zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    S. Bu, Some remarks about the R-boundedness, Chinese Ann. Math. Ser. B 25 (2004), no. 3, 421–432.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    Ph. Clément, B. de Pagter, F.A. Sukochev, and H. Witvliet, Schauder decompositions and multiplier theorems, Studia Math. 138 (2000), 135–163.zbMATHMathSciNetGoogle Scholar
  6. [6]
    Ph. Clément and J. Prüss, An operator-valued transference principle and maximal regularity on vector-valued L p-spaces, 67–87 in Evolution Equations and Their Applications in Physics and Life Sciences, Lumer and Weis (eds.), Marcel Dekker, 2000.Google Scholar
  7. [7]
    R. Denk, M. Hieber, and Jan Prüss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc. 166 (2003), no. 788.Google Scholar
  8. [8]
    M. Girardi and L. Weis, Operator-valued Fourier multiplier theorems on L p(X) and geometry of Banach spaces, J. Funct. Anal. 204 (2003), no. 2, 320–354.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    P. Halmos, Measure theory, Graduate Texts in Mathematics 18, Springer-Verlag, New York, 1974.zbMATHGoogle Scholar
  10. [10]
    M. Hoffmann, N. Kalton, and T. Kucherenko, R-bounded approximating sequences and applications to semigroups, J. Math. Anal. Appl. 294 (2004), no. 2, 373–386.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    H. Hogbe-Nlend, Théorie des Bornologies et Applications, Lecture Notes in Mathematics 213, Springer-Verlag, Berlin, 1971.zbMATHGoogle Scholar
  12. [12]
    P. Kunstmann and L. Weis, Maximal L p-regularity for parabolic equations, Fourier multiplier theorems and H -functional calculus, 65–311 in Functional analytic methods for evolution equations, Lecture Notes in Mathematics 1855, Springer-Verlag, Berlin, 2004Google Scholar
  13. [13]
    C. Le Merdy and A. Simard, A factorization property of R-bounded sets of operators on L p-spaces, Math. Nachr. 243 (2002), 146–155.zbMATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Sequence Spaces, Springer-Verlag, Berlin, 1977.zbMATHGoogle Scholar
  15. [15]
    J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Function Spaces, Springer-Verlag, Berlin, 1979.zbMATHGoogle Scholar
  16. [16]
    B. Maurey and G. Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétes géométriques des espaces de Banach, Studia Math. 58 (1976), no. 1, 45–90.zbMATHMathSciNetGoogle Scholar
  17. [17]
    J. Prüss, Maximal regularity for evolution equations in L p-spaces, Conf. Semin. Mat. Univ. Bari no. 285 (2002), 1–39 (2003).Google Scholar
  18. [18]
    L. Weis, Operator-valued Fourier multiplier theorems and maximal L p-regularity, Math. Ann. 319 (2001), 735–758.zbMATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    H. Witvliet, Unconditional Schauder decompositions and multiplier theorems, PhD-thesis, Delft University of Technology, Delft, 2000.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Onno van Gaans
    • 1
  1. 1.Institute for Stochastics Department for Mathematics and Computer ScienceFriedrich Schiller University JenaJenaGermany

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