On R-boundedness of Unions of Sets of Operators

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 τ 1 , τ 2 , . . ., 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 = 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.

R-Boundedness of Smooth Operator-Valued Functions

“…Arendt and Bu [3,Proposition 1.13] pointed out that uniform boundedness already implies R-boundedness if (and only if) X has cotype 2 and Y has type 2. Recently, van Gaans [12] showed that a countable union of R-bounded sets remains R-bounded if the individual R-bounds are ℓ r summable for an appropriate r depending on the type and cotype assumptions, improving on the trivial result with r = 1 (the triangle inequality!) valid for any Banach spaces.…”

$R$ -Boundedness versus $\gamma$ -boundedness

“…(i) For other connections between R-boundedness, type and cotype we refer to [3], [10], [12], [14] and [36]. (ii) Recall the following result due to Pisier.…”