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

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space} with \reftext{lem:vector-space-basic-identities-2026a}{zero vector} $0_{V}$, and let $T$ be a \reftext{def:linear-operator-2026a}{linear operator} on $V$ that is \reftext{def:self-adjoint-operator-2026b}{self-adjoint} and \reftext{def:positive-semidefinite-operator-2026a}{positive semi-definite}. Let $|z|$ denote the \reftext{def:complex-modulus-2026a}{modulus} of a \reftext{def:complex-numbers-2026a}{complex number} $z$. Then the following hold.

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

an inequality between \reftext{def:real-numbers-c54-2026c}{real numbers} in the order of the \reftext{def:ordered-field-c54-2026b}{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.

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