Davidson K.R. and Herrero D.A. proved that every Riesz operator T on a Banach space having F.D.p. B.D. has West decomposition, i.e. T can be decomposed as a sum of a compact operator and a quasinilpotent operator [Indiana Univ. Math. J. 35 (1986), 333-343; MR 87f: 47023]. Later the author extended their result to spaces Lp(u), 1 < p < ∞ (MR 90c: 47031). In this paper, first, the author proves that for the sequence {Xi} of Banach spaces, the Banach space (∑Xi)lp., (1 < p < ∞) is B-convex if and only if {Xi} are so called "Uniformly" B-convex, i.e. there exists an integer n ≥ 2 such that supB(n, Xi) < n where B(n, Xi) is the B-convexity constant of Xi for n. Secondly, the author proves that every Riesz operator on a B-convex space has West decomposition, applying local theory of Banach spaces. In view of the facts that every Banach space X has a type p(X), 1 ≤ p(X) ≤ 2 and that type p(X) > 1 is equivalent to X being B-convex, this result explains that further research of the problem of West decomposition of Riesa operators may be limited to the operators on type-1 spaces.
