Shift operators and their classification

The paper’s Theorem 2.10 states and proves that, on a finite-dimensional Banach space X with a basis E having S-bounded projections, the shift σ_S on ℓ^p(X) is conjugate to a finite product of bilateral weighted backward shifts. The proof constructs coordinate maps Γ_b, assembles them into an isomorphism I: ℓ^p(X) → (ℓ^p(K))^{dim X}, and shows I ∘ σ_S = (∏_{x∈E} B_{ω(x)}) ∘ I, using the identity S_{n+1}(e_{n+1}(x)) = ω_{n+1}(x) e_n(x) and the S-bounded projections hypothesis to guarantee boundedness and surjectivity of the coordinate maps . The candidate solution reproduces the same strategy (coordinate extraction along the moving normalized bases and synthesis), differing only in notation (coordinate functionals φ_{n,x} vs. projections Π_{e_n(x)}) and in showing the inverse explicitly rather than appealing to the open mapping theorem. No substantive gaps or contradictions were found.