Proof of Elementary Properties of a Self-Adjoint Operator

Conditions on an inner product are numbered as in Complex Inner Product Space, and (M$k$) denotes claim $k$ of Properties of Complex Conjugation and Modulus. Throughout we use that for $c$ a complex number and $y,z\in V$,

by conditions 1 and 3 and (M1).

Claim 1. Let $x\in V$. By condition 1 and self-adjointness,

A complex number equal to its own conjugate is real by (M1), so $\langle x,T(x)\rangle$ is a real number.

Claim 2. Let $x$ be an eigenvector of $T$ with eigenvalue $\lambda$, so $x\ne 0_{V}$ and $T(x)=\lambda x$. By condition 3,

By condition 4 the number $\langle x,x\rangle$ is real, and it is nonzero since $\langle x,x\rangle=0$ would force $x=0_{V}$. Hence $\lambda=\langle x,T(x)\rangle\,\langle x,x\rangle^{-1}$. By claim 1 the first factor is real; the second is the inverse of a nonzero real number, which is again real, and the product of two real numbers formed in the field of complex numbers is their product in $\mathbb{R}$, by condition 1 of The Complex Numbers together with claim 1 of Canonical Form and Arithmetic of Complex Numbers. Hence $\lambda$ is a real number.

Claim 3. By claim 1 of The Span of a Finite Family is the Smallest Subspace Containing It the span $U$ of $\tilde{u}$ is a linear subspace, its elements are exactly the vectors $cu$ with $c$ complex (by the definition of the span and claim 1 of Properties of Finite Sums of Vectors), and $u\in U$.

Let $x\in W$, so $\langle w,x\rangle=0$ for every $w\in U$; in particular $\langle u,x\rangle=0$. Let $cu\in U$. Using self-adjointness, then $T(u)=\lambda u$, then the identity recorded at the start,

where $T(cu)=cT(u)=c(\lambda u)=(c\lambda)u$ by linearity of $T$ and the vector space axioms of that definition. As $cu$ was an arbitrary element of $U$, the definition of the orthogonal complement gives $T(x)\in W$.

Claim 4. By claim 1 of A Linear Subspace is a Vector Space and Inherits an Inner Product the set $W$ is a complex vector space under the operations of $V$, and by claim 3 of that lemma it is a complex inner product space whose inner product is the restriction of $\langle\cdot,\cdot\rangle$. The hypothesis says that $x\mapsto T(x)$ maps $W$ into $W$, and it is additive and homogeneous there because $T$ is and because the operations of $W$ are those of $V$; hence it is a linear operator on $W$. Finally, for $x,y\in W$ the inner products of $W$ agree with those of $V$, so self-adjointness of $T$ on $V$ gives $\langle T(x),y\rangle=\langle x,T(y)\rangle$; that is, the restriction is self-adjoint on $W$.