Operator Factorization Beyond Hilbert Spaces: Banach Duality and Stable Levy Processes

We extend the operator factorization framework for stochastic calculus from Hilbert to Banach energy spaces. The operator derivative D_X, defined as the Banach dual of the stochastic integral, yields a fluctuation factorization under an explicit representation hypothesis (Theorem A). The principal result (Theorem B) is a product rule with Leibniz defect for symmetric γ-stable Lévy processes (γ ∈ (1,2)): the integrand space L^p with p < γ < 2 is non-Hilbert, and the failure of the Leibniz rule is measured by the sum of product jump defects — without reference to quadratic variation, which is infinite in this setting.