Spectral theorem for unbounded self-adjoint operators

Remarks

One proof uses the Cayley transform $U = (A - iI)(A + iI)^{-1}$, a unitary operator for which $1$ is not an eigenvalue. The spectral theorem for bounded normal operators applied to $U$ gives a projection-valued measure on the unit circle. It is carried over to $\R$ by the inverse of the map $\lambda \mapsto (\lambda - i)/(\lambda + i)$. Equivalently, if $H$ is separable, $A$ is unitarily equivalent to the multiplication by a real measurable function $\varphi$ on some $L^2(\Omega, \mu)$, on the domain of all $f$ with $\varphi f \in L^2(\Omega, \mu)$. Example: on $L^2(\R)$, the momentum operator $-i \, d/dx$ on the Sobolev space $H^1(\R)$ is self-adjoint; the Fourier transform turns it into multiplication by the variable $\xi$, with spectrum $\R$. Self-adjointness, not mere symmetry, is needed. The operator $-i \, d/dx$ on smooth functions with compact support in $(0, \infty)$ is symmetric, but it has no self-adjoint extension and no spectral resolution of this form.

Going deeper

For a student

The theorem is the mathematical basis of quantum mechanics, where observables such as position, momentum and energy are unbounded self-adjoint operators. It defines $f(A) = \int f(\lambda) \, dE(\lambda)$ for Borel functions $f$, in particular the unitary group $e^{itA}$ that gives the time evolution of a quantum system. The probability that a measurement of $A$ in the unit state $x$ gives a value in $B$ is $\ip{E(B) x}{x}$. Stone's theorem gives the converse correspondence between strongly continuous one-parameter unitary groups and self-adjoint operators.