I see in papers that people use the following
Operators are bounded, and $|A|=(A^{\ast}A)^{\frac{1}{2}}$. $$\sup_{||x||=1}\langle (|A|^{2}+|B|^{2})x,x\rangle =|||A|^{2}+|B|^{2}||$$
I am confused as, isn't $$||T||=\sup_{||x||=1}||Tx||$$ and $$||Tx||=(Tx,Tx)^{\frac{1}{2}}$$
That would mean that $$||T||=\sup_{||x||=1}(\langle Tx,Tx\rangle)^{\frac{1}{2}}$$
It looks like they are using norm definition $$||T||=\sup_{||x||=1}\langle Tx,x\rangle$$??
Thank you for helping me, I am a bit confused about the notation or something it seems.