Cauchy-Schwarz Inequality for a Positive Semi-Definite Self-Adjoint Operator

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$, and let $T$ be a linear operator on $V$ that is self-adjoint and positive semi-definite. Let $|z|$ denote the modulus of a complex number $z$. Then the following hold.

1. (Cauchy-Schwarz for the form of $T$) For all $u,v\in V$,

$$\bigl|\langle u,T(v)\rangle\bigr|^{2}\le\langle u,T(u)\rangle\,\langle v,T(v)\rangle ,$$

an inequality between real numbers in the order of the ordered field $\mathbb{R}$: the two factors on the right are nonnegative real numbers by the definition of a positive semi-definite operator, so their product is formed in $\mathbb{R}$, and the left-hand side is a square of a real number.

2. (Null vectors) If $u\in V$ satisfies $\langle u,T(u)\rangle=0$, then $T(u)=0_{V}$.