The Spectral Theorem of Unbounded Operators

Proposition

Let $(X,\mu )$ be a measure space with $\mu$ a finite measure. Suppose that $f$ is a measurable, real-valued function on $X$ which is finite a.e. Then the operator $T_f:\phi \mapsto f\phi$ on $L^2(X,\mu )$ with domain
$D(T_f)=\{\phi \mid f\phi \in L^2(X,\mu )\}$
is self-adjoint and $\sigma (T_f)$ is the essential range of $f$.

Spectral Theorem - Multiplication Operator Form

Let $T$ be a self-adjoint operator on a separable Hilbert space \H. Then there exists a measure space $(X,\mu )$ with $\mu$ finite, a unitary operator $U:\stackrel{˝}{\to }{L}^2(X,\mu )$, and a real-valued function $f$ on $X$ which is finite a.e. such that

• $\psi \in D(T)$ if and only if $x\mapsto f(x)(U\psi )(x)\in L^2(X,\mu )$;
• If $\phi \in U(D(T))$, then $(UT{U}^{-1}\phi )(x)=f(x)\phi (x)$.
  

The Functional Calculus Form

Spectral Theorem - Functional Calculus Form

Let $T$ be a self-adjoint operator on a separable Hilbert space \H. Then there exists a unique map $\Phi :B_b(\mathbb{R})\to B(\stackrel{˝}{)}$ such that

• $\Phi$ is a $\ast$-homomorphism;
• $\Phi$ is norm continuous. i.e., $\Vert \Phi (h){\Vert }_{B(\stackrel{˝}{)}}\le \Vert h{\Vert }_{\infty }$;
• If $\{h_n{\}}_{n=0}^{\infty }\subset B_b(\mathbb{R})$ is a sequence of bounded Borel functions with $h_n(x)\to x$ for each $x$ and $|h_n(x)|\le |x|$ for all $x$ and $n$. Then, for any $\psi \in D(T)$, ${\lim }_{n\to \infty }\Phi (h_n)\psi =T\psi$.
• If $h_n(x)\to h(x)$ pointwise and if the sequence $\Vert h_n{\Vert }_{\infty }$ is bounded, then $\Phi (h_n)\to \Phi (h)$ strongly.
  

e.g. If $A$ is an unbounded positive self-adjoint operator on \H. Then

The Projection Valued Measure Form

Projection Valued Measure

Let $(\mathbb{R},{\mathcal{B}}_{{\mathbb{R}}^d})$ be the Borel measurable space. A projection-valued measure on a separable Hilbert space is a map $P:\mathcal{B}(\mathbb{R})\to B(\stackrel{˝}{)}$ such that $P(\Omega )$ is an orthogonal projection, $P(\varnothing )=0$, $P(\mathbb{R})=I$, and for any countable collection $\{{\Omega }_i{\}}_{i=1}^{\infty }$ of disjoint Borel sets, we have $P\left({\bigcup }_{i=1}^{\infty }{\Omega }_i\right)={\sum }_{i=1}^{\infty }P({\Omega }_i),$where the sum converges in the strong operator topology.

Spectral Theorem - Projection Valued Measure Form

There exists a one-to-one correspondence between self-adjoint operators on a separable Hilbert space \H and projection valued measures $\{P_{\Omega }\}$ on \H: $T={\int }_{\mathbb{R}}\lambda \text{dd}{P}_{\lambda }$