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 DX, 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 Lᵖ 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.