I'm studying fractional powers of an operator and came across an exercise I could not solve.
Let $A: D(A)\subset L^2(0,\pi) \to L^2(0,\pi)$ be a linear operator defined by \begin{gather*} D(A) = H^2(0,\pi) \cap H_0^1(0,\pi) \\ Au = -u_{xx} \end{gather*}
How do I prove that $A^{\frac{1}{2}}u = iu_x$?
A bit of context: I am actually trying to prove that \begin{equation*} \|u_x\|_{L^2(0,\pi)} \leq \|Au\|_{L^2(0,\pi)}^{\frac{1}{2}}\|u\|_{L^2(0,\pi)}^{\frac{1}{2}} \end{equation*}
I do know that this will fairly simply come from the Momentum Inequality \begin{equation} \|A^\alpha u\|_{L^2(0,\pi)} \leq K\|Au\|_{L^2(0,\pi)}^\alpha\|u\|_{L^2(0,\pi)}^{1-\alpha} \quad,\quad 0 \leq \alpha \leq 1 \end{equation} but I am struggling to prove that $A^{\frac{1}{2}}u = iu_x$. Intuitively, I know it should be true, but the calculations are not leading me anywhere. Perhaps I am missing something basic here?