Operators

We begin by defining self-adjoint and positive operators.

Definition: Let $A:H \longrightarrow H$ be an operator, then $A$ is said to be:

1. Self-adjoint if $\langle Af,g \rangle=\langle f,Ag \rangle$, $\forall f \in H$, $\forall g \in H$.

2. Positive if $\langle Af,f \rangle \geq 0$, $\forall f \in H$. If $A$ is positive, then we write $A \geq 0$.

3. Normal if $A^{*}A=AA^{*}$.

4. Coercive (or bounded away from zero by $\gamma$) if there exists a positive constant $\gamma$ such that
   $\Vert Af\Vert \geq \gamma \Vert f\Vert$, $\forall$ $0 \neq f \in H$.

5. Unitary if it is surjective and $\langle Ux,Uy \rangle = \langle x,y \rangle$, for all $x,y \in H$.

Definition: For any function $f$ and any real number $a$, the dilation, translation, and modulation operators are defined, respectively, as

$D_af(x)=\frac{1}{\sqrt{\vert a\vert}}f(\frac{x}{a})$, $a \neq 0$

$T_af(x)=f(x-a)$

$E_af(x)=e^{2 \pi i a x} f(x)$.

It is easy to prove that the dilation, translation, and modulation operators are unitary from $L^2(\mathbb{R})$ onto $L^2(\mathbb{R})$.