On R-boundedness of unions of sets of operators

It is shown that the union of a sequence T1, T2, . . . of R-bounded sets of operators from X into Y with R-bounds τ1, τ2, . . ., respectively, is Rbounded if X is a Banach space of cotype q, Y a Banach space of type p, and ∑ ∞ k=1 τ r k <∞, where r = pq/(q−p) if q <∞ and r = p if q =∞. Here 1 ≤ p ≤ 2 ≤ q ≤ ∞ and p 6= q. The power r is sharp. Each Banach space that contains an isomorphic copy of c0 admits operators T1, T2, . . . such that ‖Tk‖ = 1/k, k ∈ N, and {T1, T2, . . .} is not R-bounded. Further it is shown that the set of positive linear contractions in an infinite dimensional L is R-bounded only if p = 2.