My question : Suppose T is a normal operator on a Hilbert space H. Show that there exists a self-adjoint operator S on H such that T=f(S), where f is continuous function from spectrum of S into S.
My approach to the problem was to find out an operator S and show that the $C^*$ algebra generated by $I$ and $S$ includes T. Then, continuous functional calculus would prove the result. So my first guess was to choose $S=(T+T^*)/2$. However, I have been unable to show that T is the limit of some polynomial in S.
Is my intuition about the problem correct? In that case, please provide some suggestions regarding how to show that T is in the unital sub-algebra generated by S?
Idea: $\mathbb{R}^m\cong\mathbb{R}^n$
Choose function:* $$\eta\in\mathcal{B}(\mathbb{C},\mathbb{R}):\quad \vartheta\circ\eta=\mathrm{id}$$ Construct operator: $$S:=\eta(N):\quad S=S^*$$ By Borel calculus: $$N=\mathrm{id}(N)=\vartheta(\eta(N))=\vartheta(S)$$ *Note discontinuity!