Limit of a sequence in the space $\ell_2$ I have difficulties in the following problem. 
Let $H=\ell_2$ be the space of square-summable sequences. Let $\alpha\in (0,1)$ and $\{u^k\}\subset H$ be such that
$$
u^{k+1}=(1-\alpha)u^k+\alpha T(u^k),
$$
where
$$
T(u)=(0, u_1, u_2, u_3, \ldots,u_n, \ldots)
$$
for all $u=(u_1, u_2, u_3, \ldots,u_n, \ldots)\in H$. Choose $u^0\in H$ and $\alpha\in (0, 1)$ such that $\{u^k\}$ converges weakly to $0$ but does not converge strongly to $0$.
Thank you for all hints and comments.
 A: It seems the following.
$$u^k=\sum_{i=0}^k \binom ki (1-\alpha)^i\alpha^{k-i} T^i(u^0).$$
Then for each $a\in\ell_2$ we have 
$$(u^k,a)=\sum_{i=0}^k \binom ki (1-\alpha)^i\alpha^{k-i} (T^i(u^0),a)=
\sum_{i=0}^k \binom ki (1-\alpha)^i\alpha^{k-i} (u^0, T^{*i}(a)),$$
where $T^*(a_1,a_2,a_3,\dots)= (a_2,a_3,a_4\dots)$. 
Now let $\alpha\in (0,1)$ and $a\in\ell_2$. We are going to show the weak convergence of the sequence $\{u^k\}$ to $0$. Let $\varepsilon>0$ be arbitrary. Since $\lim_{i\to\infty} ||T^{*i}(a)||^2=\lim_{i\to\infty} \sum_{n=i}^\infty a_n^2=0$ there exists a number $N$ such that $||T^{*i}(a)||<\varepsilon$ for each $i>N$. Since $\alpha\in (0,1)$ there should be a number $K\ge N$ such that $\sum_{i=0}^N \binom ki (1-\alpha)^i\alpha^{k-i}<\varepsilon$ for each $k>K$. Then for each $k>K$  we have 
$$(u^k,a)=\sum_{i=0}^k \binom ki (1-\alpha)^i\alpha^{k-i} (u^0, T^{*i}(a))=
\sum_{i=0}^N \binom ki (1-\alpha)^i\alpha^{k-i} (u^0, T^{*i}(a))+
\sum_{i=N+1}^K \binom ki (1-\alpha)^i\alpha^{k-i} (u^0, T^{*i}(a))\le
\varepsilon\max_i ||u^0||||T^{*i}(a)||+ \sum_{i=N+1}^K \binom ki (1-\alpha)^i\alpha^{k-i}\varepsilon\le \varepsilon||u^0||||a||+\varepsilon.$$
--
I suspect that for $u^0=(1,1/2,1/3,1/4,\dots)$ the sequence $\{u^k\}$ does not converge strongly to $0$, but now I see no simple way to do respective estimations.
