Dilation operator

The dilation operator $$D_c$$ is defined on $$L^(\mathbb R)$$ (see Def. 6.2.1 ; not sure if it can be defined for other $$L^p(\mathbb R)$$ spaces too). By definition, for a given $$c>0$$ this operator acts as:

$$(D_c f)(x):=\frac{1}{\sqrt{c}} f(\frac{x}{c}), \,x\in \mathbb R.$$

$$D_c$$ is a unitary operator and it holds that
 * $$D_c^{-1} = D_{1/c} = (D_c)^*$$.