Normal bounded operator

Question

Let $T$ be a bounded normal operator on a Hilbert space.

Now I have to show that $T$ is self-adjoint if and only if $\sigma(T) \subset \mathbb{R}$.

I already know that for an Abelian unital C-star-algebra $A$, with a generating element $a\in A$; $A=\text{Alg}_{C^*}(e,a)$, it holds that $a$ must be normal and that $a$ is self-adjoint if and only if $\sigma(a)\subset \mathbb{R}$.

But I don't know how I parse to bounde operators? Anyhelp with this?

Answer 1

You can take $A$ to be the C$^*$-algebra generated by $T$ and then use the result you mention.