Operator for scaling a function? Let $\mathbb{F}$ denote the set of functions of the form $f: \mathbb{R} \to \mathbb{R}$. 
I am interested to know whether there exists a well-known linear map $T_\alpha: \mathbb{F} \to \mathbb{F}$ such that $Tf(x) = f(\alpha x)$.
What would be the formal name of such an operator?

It is called the dilatation operator, and it is but a coordinate change of the Lagrange shift operator.    Define 
$$
x\equiv e^y , \qquad \alpha \equiv e^\beta,
$$
so that 
$$
T f(e^y)= f(e^{y+\beta}).
$$
That is, T is a shifter,
$$
T= e^{\beta \frac{d}{dy}}=e^{\ln\! \alpha ~~~x\frac{d}{dx}},
$$
so that 
$$
T f(x) = f(\alpha x).
$$
