Operator Relations Characterizing Derivatives

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This monograph develops an operator viewpoint for functional equations in classical function spaces, addressing a gap in the mathematical literature. Major operations in analysis are often defined by elementary properties or equations they satisfy. The authors explore how the derivative is characterized by equations like the Leibniz rule or the Chain rule in Ck-spaces. Through localization, these operator equations transform into specific functional equations, which the authors solve. The study of second-order operator equations is motivated by the second derivative, Sturm-Liouville operators, and the Laplacian. The authors provide general solutions to these operator equations under weak non-degeneration assumptions, without requiring linearity or continuity conditions. They find that the Leibniz rule, the Chain rule, and their extensions remain stable under perturbations and relaxed assumptions. This work offers an algebraic understanding of first- and second-order differential operators, revealing the rich operator-type structure behind the derivative and its related concepts in analysis. The book is accessible to a general mathematical audience, as it does not assume prior knowledge of functional equations and includes all necessary results and proofs.