# Lecture 18: Unbounded Spectral Theorem
###### tags: `224a 2020`

We begin by stating a very concrete criterion for checking selfadjointness which how one checks this property in practice.
**Theorem.** If $T:D(T)\rightarrow H$ is symmetric, the following are equivalent:
1. $T$ is self-adjoint.
2. $ran(T\pm i)=H$.
3. $T$ is closed and $ker(T\pm i)=\{0\}.$

*Proof.* Dimock 1.3.2:thm1.9 for parts 1 and 2. For the full theorem, see Reed and Simon VIII.3$\square$

## Spectral theorem
$\newcommand{\C}{\mathbb{C}}$
Let $A:D(A)\rightarrow A$ be selfadjoint. The key observation is that the characterization in the previous lecture implies $ran(A+i)=H$, which means that $C_A:=(A+i)^{-1}\in L(H)$. We further observe $C_A$ is unitary and therefore normal. Therefore, by the spectral theorem for *normal* operators, there is a unitary $U:H\rightarrow L^2(X,\mu)$ and a bounded multiplication operator $M_h$ with $g:X\rightarrow \C$ such that
$$ U^{-1}C_AU = M_h\quad on L^2(X,\mu).$$
The change of variables $g=h^{-1}-i$ yields $UAU^{-1}=M_g$ on $L^2(X,\mu)$, after verifying that $h\neq 0$ a.e. and $g$ is real.

The full proof is given in Reed and Simon VII.3.