$\|A\|^2=\|A^* A\|$ for linear bounded operators on hilbert space?

Question

I'm reading some functional analysis notes and the author use the fact that for bounded linear operator $A$ on Hilbert space holds $\|A\|^2=\|A^*A\|$.
I know that holds $\|A^*A\|\leq\|A^*\|\|A\|=\|A\|\|A\|$ but I'm having difficulties proving the other inequality.
I really appreciate any kind of help.

Answer 1

Use the fact that $A^*A$ is self-adjoint $$ \|A^*A\|=\sup_{\|u\|=1}|(A^*Au,u)|=\sup_{\|u\|=1}|(Au,Au)|=\sup_{\|u\|=1}\|Au\|^2=\|A\|^2 $$