Transform Models, Semigroup Generation, and Optimal Weights for Hybrid Fractional Operator Algebras

We develop a unified analytic framework for the continuous and discrete Banach completions of hybrid fractional operator algebras. The point of view is coefficient-theoretic: for a spectral index set Λ and one-sided grade lattice N₀ʳ, we study weighted Banach spaces \(Xωᵖ = \{u=∑ ak,λek,λ: ∑ |ak,λ|ᵖω(k,λ)ᵖ<∞\}\) carrying the canonical forward shifts Jᵢ, backward shifts Cᵢ, and diagonal spectral multipliers Mσ. This abstract realization simultaneously models the continuous completions built from monomial--exponential bases and the discrete completions built from factorial--character bases. We prove exact norm formulae for the one-sided shifts and show that admissibility of the hybrid algebra is equivalent to a simple ratio condition on the weight. We then establish a sharp optimal-weight theorem: for every coordinate, \(\|J_i\|\,\|C_i\|≥ 1,\) and equality for all i holds precisely for geometric weights \(ω(k,λ)=η(λ)ρ_1k_1⋯ρ_rk_r.\) Thus the geometric weights are exactly the balanced weights of the hybrid shift algebra. For geometric weights we construct a unified transform model on a polydisk. Under this transform, Jᵢ becomes multiplication by zᵢ, Cᵢ becomes the backward quotient operator, and every spectral multiplier remains pointwise diagonal in the spectral label. We further show that every maximal diagonal multiplier Mσ is closed and densely defined, and that it generates a C₀-semigroup if and only if λσ(λ)<∞, in which case the semigroup is exactly M_etσ. Finally, for mixed generators \(Aσ,a,b = Mσ+∑ a_iJ_i+∑ b_iC_i,\) we obtain C₀-generation, explicit growth bounds, and a sharp optimization principle for the geometric parameters ρᵢ. These results place the continuous and discrete completion theories into a common transform--semigroup framework and prepare the subsequent study of boundary trace ideals.