A multiplication operator is always normal, but rarely self-adjoint or unitary

Let X be the space of all continuous complex valued functions on $[0,1]$ with inner product $\langle f,g\rangle=\int_0^1f(t)\overline{g(t)}\,dt$. Let $T:X\to X$ be defined by $T(f)(t)=te^{it}f(t)$. Then: