Translation semigroups Consider the operators $Q(t)\in \mathcal{B}(L^{2}(\mathbb{R}))$ given by $(Q(t)f)(s)=f(s+t)\quad \forall f\in L^{2}(\mathbb{R})$.
Each $Q(t)$ is clearly unitary and the following are satisfied:


*

*$Q(0)=$Identity on $L^{2}(\mathbb{R})$.

*$Q(s+t)=Q(s)Q(t)$
We now must show that $\lim_{t\to0}\|Q(t)f-f\|=0\quad\forall f\in L^{2}(\mathbb{R})$.
I'd be grateful for some hints on how to prove this, as always!

You have to make use of this lemma : 
Let $T(\cdot) : R_+ → B(X)$ semigroup of operators. Then the following assertions are equivalent.
(a) $T(\cdot)$ is strongly continuous (and thus a $C_0$–semigroup).
(b) It holds $\lim_{t \rightarrow 0_+} T(t)x = x$ for all $x \in X$.
(c) There are a number $t_0 > 0$ and a dense subspace $D \subseteq X$ such that
$\sup_{0\leq t \leq t_0} \|T(t)\| < \infty$ and $\lim_{t \rightarrow 0_+} T(t)x = x$ for all $x \in D$.
Take $D = C_c^{\infty}(\mathbb{R})$ i.e. compactly supported smooth functions on $\mathbb{R}$, and $X = L^2(\mathbb{R})$, and the semigroup is translation semigroup in question. 
HINT : Utilise (c).   
