Prove that $U_{\alpha}$ is a family of unitary operators We are in the complex Hilbert space $l^2(\mathbb{N})$ and we have a family of operators, $\alpha \in\mathbb{R} $:
$$U_{\alpha}: \{c_n\}_{n=1}^\infty\to\{e^{-in\alpha}c_n\}_{n=1}^\infty$$
1) Prove that $U_{\alpha}$ are unitary operators
2) Prove that, for $\alpha=0$ , they satisfy the property of strong continuity and the property of being an abelian group.
1) I know that in $l^2(\mathbb{N})$ the inner product between $c=\{\gamma_n\}_{n=1}^\infty$ and $b=\{\beta_n\}_{n=1}^\infty$ is defined as it follows:
$$(c|b)=\sum_{k=1}^\infty\overline{\gamma_k}\beta_k$$
I know that, to be an unitary operator:
$$(Uc|Ub)=(c|b)$$
So
$$(c|b)=\sum_{k=1}^\infty\overline{\gamma_k}\beta_k=(Uc|Ub)=\sum_{k=1}^\infty\overline{\gamma_k e^{-ik\alpha}}\beta_k e^{-ik\alpha}=\sum_{k=1}^\infty\overline{\gamma_k}\beta_k e^{+ik\alpha}e^{-ik\alpha}=(c|b)$$
Right?
2) I don't know how to approach this further requirement.
 A: For 1): from the definition of $U_\alpha$ it is obvious that $U_\alpha=\operatorname{diag}(e^{-i\alpha},e^{-2i\alpha},e^{-3i\alpha},\ldots)$ (verify that $\langle e_i,U_\alpha e_j\rangle=0\ \forall i\neq j$ so the non-diagonal matrix elements vanish; here $(e_j)_{j\in\mathbb N}$ is the standard basis of $\ell_2$). As you now have a matrix form you can just check that $U_\alpha U_\alpha^\dagger=U_\alpha^\dagger U_\alpha=\operatorname{id}_{\ell_2}$ so $U_\alpha$ is unitary as it evidently is a linear bounded operator in the first place.
2) The family of operators $(U_\alpha)_{\alpha\in\mathbb R}$ is strongly continous in $\alpha=0$ if
$$
\lim_{\alpha\to 0}\Vert (U_\alpha-U_0)x\Vert=\lim_{\alpha\to 0}\Vert (U_\alpha-\operatorname{id}_{\ell_2})x\Vert=0
$$
for all $x\in\ell_2$ where $\Vert\cdot\Vert=\sqrt{\langle \cdot,\cdot\rangle}$ is the norm induced by the scalar product on $\ell_2$. Also with the explicit matrix form of the $U_\alpha$ it is easy so show that they form a group which additionally is abelian. Hint: it might be helpful to verify that $U_\alpha U_\beta=U_{\alpha+\beta}$ for any $\alpha,\beta\in\mathbb R$.
Hope this gives you the right ideas on how to proceed!
