Since this was asked in another chatroom and it was very likely answered on the site before, I thought it might be a good exercise in searching for questions.

I was able to find 1196237: Show that in a complex Hilbert space, T normal bounded linear operator, $\| T^2 \| =\| T \| ^2$.

I tried Google: norm normal "T^2" site:math.stackexchange.com. It was among the top hits.

This question also comes up among questions tagged normal-operator+norm. (But this is cheating, since I have retagged it recently.)

When searching in Approach0 for $\|T^2\| = \|T\|^2$, the question is among the top results, too.

Are there some other posts about the same question which I missed? Are there some other tricks I should have tried when searching?