Cauchy-Schwarz inequality for a positive operator ($|⟨Tx,y⟩{|}^2\le ⟨Tx,x⟩⟨Ty,y⟩$)

For a bounded positive operator $T$, the form $B(x,y)=⟨Tx,y⟩$ is positive semidefinite and Hermitian. Expanding $B(x+ty,x+ty)\ge 0$ for complex $t$ and minimizing its quadratic polynomial gives the displayed inequality. If $B(y,y)=0$, varying $t$ along the negative direction of $B(x,y)$ forces $B(x,y)=0$, covering the degenerate case. In particular $⟨Tx,x⟩=0$ implies $Tx=0$, by testing against arbitrary $y$.