My question is about a linear map defined on the set of smooth periodic functions.
Precisely, let $C$ be the set of infinitely many times continuously differentiable and $2\pi$ periodic functions, whose domain is restricted to $[0,2\pi]$. Let $T:C\to C$ be a linear map. Assume $\frac{d}{dx}(Tf)=T(\frac{df}{dx})$ and $T(e^{ix}f(x))=e^{ix}(Tf)(x)$ for any $x\in [0,2\pi]$ and for any $f\in C$ . My conjecture is $T$ must be the identify function, i.e. $Tf=f$ for any $f\in C$. Does anyone have idea for the proof?