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)


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.


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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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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