Basisity problem and weighted shift operators

We investigate a basisity problem in the space and in its invariant subspaces. Namely, let W denote a unilateral weighted shift operator acting in the space , 1≤ p∞, by Wzn=λnzn+1,n≥0, with respect to the standard basis {zn}n≥0. Applying the so-called “discrete Duhamel product” technique, it is proven that for any integer k≥1 the sequence {(wi+nk)−1(W|Ei)knf}n≥0 is a basic sequence in Ei:=span {zi+n:n≥0} equivalent to the basis {zi+n}n≥0 if and only if . We also investigate a Banach algebra structure for the subspaces Ei,i≥0.