Normal operators, verification

Question

I have been reading some article and I came across this equality $$\big\| |T|+|T^*|\big\|^2 = \big\|(\,|T|+|T^{*}|\,)^2\big\|\,,$$ where $|T|=(T^{*}T)^{\frac12}$. The only way this can hold is if the operator under the norm is a normal operator, that is I believe for normal operators it holds that $$\|T\|^{n}=\|T^n\|$$ The author didn't state this, or even mention it, but I believe it to be the case. I mean, if $T_1=|T|+|T^*|$, then it holds that $T_1^*=T_1$. This holds because $$|T|^*=|T|$$ and $|T^*|^*=|T^*|$. Then checking the condition whether $T_1$ is normal, we get $$T_{1}^{*}T_{1}=T_{1}T_{1}^{*}$$ and it is the same, therefore $T_1$ is normal and we can use the fact $$\|T^n\|=\|T\|^n\,.$$ Am I correct?

Answer 1

The author possibly did not state or mention it, because the bounded linear operators on a Hilbert space form a $C^*$-algebra, in which you always enjoy the $C^*$-property: $\:$this is the equality $$\|a^*a\| = \|a\|^2\,\;\forall\, a\,.$$ Thus, $\|a^2\| = \|a\|^2,\;$ whenever $a$ is self-adjoint.

$|T|$, and $|T^*|$, and their sum $|T|+|T^*\,|$ are self-adjoint operators. Which are normal operators in particular, so yes, you are correct.