On R-boundedness of Unions of Sets of Operators

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 r = pq/(q − p) if q r = p if q = ∞. Here 1 ≤ p ≤ 2 ≤ q ≤ ∞ and p ≠ q. 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.