Unbounded Operators

By the Helinger-Toeplitz Theorem, we know that the Consider $L^2(\mathbb{R})$ and let $D(T)$ be the set of all functions $f\in L^2(\mathbb{R})$ such that ${\int }_{\mathbb{R}}{x}^2|f(x){|}^2\text{dd}x<\infty$. For $f\in D(T)$ define an operator $T$ by $(Tf)(x):=xf(x).$Then $T$ is clearly unbounded since if

Domain of an Operator

Closed Operator

An operator T\colon \H_{1}\to \H_{2} between two Hilbert spaces is called closed if its graph is closed in \H_{1}\times \H_{2}.

Extension

Let T_{1},T_{2}\colon \H_{1}\to \H_{2} be operators between two Hilbert spaces. We say that $T_1$ is an extension of $T_2$ if their graphs satisfy $\Gamma (T_1)\supset \Gamma (T_2)$ and we write $T_1\supset T_2$.

Proposition

$T_1\supset T_2$ if and only if $D(T_1)\supset D(T_2)$ and $T_{1}f=T_{2}f$ for all $f\in D(T_2)$.

Proof

Closable

An operator

Stone's Theorem

Let $U(t)$ be a strongly continuous one-parameter unitary group on a Hilbert space \newcommand{\H}{\mathcal{H}}\H. Then, there is a self-adjoint operator $H$ on \H so that $U(t)=e^{itH}$ for all $t\in \mathbb{R}$.