Is the differentiation operator normal?

Question

In the space of polynomials (degree not higher than n) , the scalar product is given by the formula: $$(f,g)=\int_{0}^{1}f(x)g(x)dx$$

is the differentiation operator normal?

Answer 1

This is never true for $n\geq 1$:

Let $D$ be the differential operator and $D^*$ it's adjoint. Then $$\langle x,D^*(1)\rangle =\langle D(x),1\rangle=\langle 1,1\rangle=1$$ so $D^*(1)\neq 0$ and hence

$$\langle DD^*(1),1\rangle=\langle D^*(1),D^*(1)\rangle=\|D^*(1)\|^2\neq 0$$ which then implies $DD^*(1)\neq 0$.

But $D^*D(1)=D^*(0)=0$, so $D^*D\neq DD^*$.