The Chain Rule Operator Equation for Polynomials and Entire Functions

Abstract

After a short survey on operator functional equations we study the solutions of the chain rule operator equation $$\displaystyle{T(f \circ g) = (Tf) \circ g \cdot Tg}$$ for maps T on spaces of polynomials and entire functions. In comparison with operators on C k -spaces with \(k \in \mathbb{N} \cup \{\infty \}\), which were studied before, there are entirely different solutions. However, these yield discontinuous maps T. Under a mild continuity assumption on T we determine all solutions of the chain rule equation on spaces of real or complex polynomials or analytic functions. The normalization condition T(−2 Id) = −2 yields Tf = f′ as the only solution.