Elementary Properties of a 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}. Then the following hold.

\textbf{1. (Real values on the diagonal)} For every $x\in V$ the \reftext{def:complex-numbers-2026a}{complex number} $\langle x,T(x)\rangle$ is a \reftext{def:real-numbers-c54-2026c}{real number}.

\textbf{2. (Real eigenvalues)} If $\lambda$ is a complex number and $x\in V$ is an \reftext{def:eigenvector-eigenvalue-2026a}{eigenvector of $T$ with eigenvalue $\lambda$}, then $\lambda$ is a real number.

\textbf{3. (Invariance of an orthogonal complement)} Let $u\in V$ be a \reftext{def:unit-vector-2026a}{unit vector} that is an eigenvector of $T$ with eigenvalue $\lambda$ for some complex number $\lambda$, let $\tilde{u}\in V^{1}$ be the \reftext{def:finite-tuple-power-2026a}{$1$-tuple} with component $u$, and let $W$ be the \reftext{def:orthogonal-complement-2026a}{orthogonal complement} of the \reftext{def:span-finite-family-2026b}{span} of $\tilde{u}$, which is a \reftext{def:linear-subspace-2026a}{linear subspace} of $V$ by claim 1 of \ref{lem:span-is-subspace-2026b} and \ref{lem:orthogonal-complement-is-subspace-2026a}. Then $T(x)\in W$ for every $x\in W$.

\textbf{4. (Restriction to an invariant subspace)} Let $W$ be a linear subspace of $V$ with $T(x)\in W$ for every $x\in W$. Then the map sending each $x\in W$ to $T(x)$ is a self-adjoint linear operator on $W$, the latter being a complex inner product space by claims 1 and 3 of \ref{lem:subspace-inner-product-space-2026b}.