On Banach Spaces with Bases

We show that a Banach spaceXhas a basis provided there are bounded linear finite rank operatorsRn: X→Xsuch that limn Rnx=xfor allx∈X,RmRn=Rmin(m, n)ifm≠n, andRn−Rn−1factors uniformly throughlmnpʹs for somep. As an application we obtain conditions on a subsetΛ⊂Zsuch thatCΛ=closed span{zk: k∈Λ}⊂C(T) andLΛ=closed span{zk: k∈Λ}⊂L1(T) have bases.