Banach Completions of Hybrid Shift-Spectral Algebra

Let \(Ωr,s:=(0,∞)^r^s,\) \({α}=(α_1,,α_r)∈(0,1)^r,\) \(Λ⊂(C_+)^s,\) and let \(ek,λ(x,y) := (∏ᵢ₌₁^r{x_ik_iα_i}{Γ(k_iα_i+1)})eλ,y\) \((k_0^r,\ λ∈Λ)\) be the canonical hybrid basis introduced in the preceding paper on mixed regions. 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 for the resulting completion. Under a natural shift-admissibility condition on ω, the partial Riemann--Liouville integrals and Caputo derivatives extend to bounded operators on Xωᵖ 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 Weyl block, while arbitrary scalar symbols define closed densely defined operators on their maximal domains. In particular, standard multidimensional Weyl operators become closed diagonal operators on Xωᵖ. For geometric weights we further prove that hybrid Mittag--Leffler series become genuine vectors of the completion. These vectors 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 \(Mσ_0+∑0<m≤ MCᵐM_{σₘ},\) under a uniform ellipticity and smallness condition, and we construct a fiberwise holomorphic model on a polydisk for geometric completions.