On Banach spaces with bases

We show that a Banach space X has a basis provided there are bounded linear finite rank operators R(n) : X --> X such that lim(n) R(n)x = x for all x is an element of X, R(m)R(n) = R(min(m, n)) if m not equal n, and R(n) - R(n-1) factors uniformly through l(p)(mn)'s for some p. As an application we obtain conditions on a subset Lambda subset of Z such that C-Lambda = closed span{z(k): k is an element of Lambda} subset of C(T) and L(Lambda) = closed span{z(k) : k is an element of Lambda} subset of L(1)(T) have bases. (C) 1996 Academic Press, Inc.