Fractional Shift Algebra on a Canonical α-Graded Space

Let 0 < α < 1, and define eₙ(x):=xnα/Γ(nα+1) for n≥ 0. We prove that the algebraic direct sum Gαᵃˡᵍ:=ₙ₌₀∞Ceₙ is the distinguished α-graded monomial space on which the order-α Riemann--Liouville integral J_α:=₀ Iₓ^α and the order-α Caputo derivative C_α:=₀CDₓ^α act as a unilateral shift pair, namely J_α eₙ=eₙ₊₁ for n≥ 0, C_α e₀=0, and C_α eₙ=eₙ₋₁ for n≥ 1. It follows that C_α J_α=I, J_α C_α=I-Π₀, and [C_α,J_α]=Π₀, where Π₀ denotes the projection onto the vacuum component. We further establish a uniqueness theorem: among graded monomial chains with one-dimensional homogeneous components, Gαᵃˡᵍ is, up to multiplication of the entire basis by a single nonzero scalar, the unique chain on which J_α and C_α act as forward and backward shifts with vacuum annihilation. Finally, for every m, we show that J_αᵐ=₀ Iₓmα on all of Gαᵃˡᵍ, whereas C_αᵐ=₀CDₓmα holds on the tail subspace Gα(≥ m):=ₙ₌ₘ∞Ceₙ. Hence the failure of the full semigroup property for Caputo derivatives is localized precisely in a finite-dimensional low-grade defect sector.