Banach Completions and Z-Transform Model for Discrete Hybrid Shift-Spectral Algebra

Let \(Ωr,sᵈⁱˢᶜ:=N_0^r^s,\) \({α}=(α_1,,α_r)∈(0,1)^r,\) \(Λ⊂(E)^s,\) \(E :=\{z:|z|>1\},\) and let \(ek,λ(n,m) := (∏ᵢ₌₁^r hk_iα_i(n_i))λ^m\) \((k_0^r,\ λ∈Λ)\) be the canonical discrete hybrid basis introduced in the preceding paper. In the present paper we pass from the algebraic direct sum generated by this basis to weighted Banach completions. For 1≤ p<∞ and a positive weight ω on N₀ʳ×Λ, we define the coefficient norm \(\|∑ ak,λek,λ\|p,ω := (∑ |ak,λ|^pω(k,λ)^p)1/p\) on the finite-support core and write Xωp,disc for the resulting completion. Under a natural shift-admissibility condition on ω, the left nabla fractional sums and Caputo nabla fractional differences extend to bounded operators on Xωp,disc and satisfy the same unilateral identities as in the algebraic model, \(C_iJ_i=I,\) \(J_iC_i=I-Π_i,\) as well as their higher-order analogues \(CᵐJᵐ=I,\) \(JᵐCᵐ=I-Π<m.\) Bounded spectral multipliers yield a diagonal functional calculus for the discrete Weyl block, while arbitrary scalar symbols define closed densely defined operators on their maximal domains. In particular, the standard multidimensional discrete Weyl fractional differences become closed diagonal operators on Xωp,disc. For geometric weights we further prove that the formal generating vectors \(E_{{ζ},λ}ᵈⁱˢᶜ := ∑k_0^r{ζ}ᵏek,λ\) become genuine vectors of the completion whenever |ζᵢ|<ρᵢ⁻¹ for all i. These are joint eigenvectors of the commuting Caputo tuple and of every spectral multiplier. Finally, we establish a Banach-space inversion theorem for mixed constant-coefficient operators of the form \(M_{σ₀} + ∑0<m≤ MCᵐM_{σₘ},\) under a uniform ellipticity and smallness condition, and we construct a fiberwise holomorphic Z-transform model on a polydisk for geometric completions.