Translation operator

The translation operator 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 $$a $$ this operator acts as:

$$(T_a f)(x):=f(x-a), \,x\in \mathbb R.$$

$$T_a$$ is a unitary operator and it holds that
 * $$T_a^{-1} = T_{-a} = (T_a)^*$$.