On Banach spaces with unconditional bases

Let X be a Banach space with a sequence of linear, bounded finite rank operators R-n : X --> X such that RnRm = R-min(n,R- m) if n not equal m and lim(n-->infinity), R(n)x = x for all x is an element of X. We prove that, if R-n - Rn-1 factors uniformly through some lp and satisfies a certain additional symmetry condition, then X has an unconditional basis. As an application we study conditions on Lambda subset of Z such that L-Lambda = closed span {z(k) : k is an element of Lambda) subset of L-1(T), where T = {z is an element of C : \z\ = 1}, has an unconditional basis. Examples include the Hardy space H-1 = Lz+.