Prove that the limit exist II

First question was here. I add one new condition.

Let $T : H \rightarrow H$ is a linear continuous unitary ($T^*=T^{-1}$) operator, $H$ is a Hilbert space (not necessary). Suppose that

1. $\forall h \in H \Rightarrow Th=h$.
2. $\forall n \in \mathbb{N} \ || T_n ||\leq 1$. NEW CONDITION
3. $T_n$ - a sequence of linear operators $T_n \underset{n\rightarrow \infty}{\longrightarrow} T$, $|| T_n - T||\rightarrow 0$. So the sequence $T_n$ tends to $T$ in the norm and we have $\forall h \in H \ \ || T_n h - T h|| \leq || T_n - T|| \ ||h||\rightarrow 0$.

Can we prove that there exists a limit of the sum $S_n= \frac{1}{n}\left( T_1 h + T_1 T_2 h + \dots + T_1 \dots T_n h \right)$ where $n \rightarrow \infty$?

• So 1 is saying that $T=Id$, right...? Then why $T$? Jun 9 '13 at 23:53
• Because in my homework $Th=h$ only in some subspace not in entire space H. I'm asking only about this particular case.
– user59928
Jun 10 '13 at 12:57

Assume that $H$ is a complex Hilbert space and define $T_k=e^{i\alpha_k} Id$, where $(\alpha_k)$ is a sequence of real numbers tending to $0$. Then all $T_k$'s are unitary and $\Vert T_k-Id\Vert\to 0$. On the other hand, we have $$S_n=\frac1n\left(e^{i A_1}+\dots +e^{i A_n} \right) Id ,$$ where $A_k=\alpha_1+\dots +\alpha_k$.
So one just needs to choose $(\alpha_k)$ in such a way that the sequence $(e^{iA_k})$ is not convergent in the Cesaro sense.
For example, one can proceed as follows (this is probably not the simplest way to do so). Let $(I_p)$ be a sequence of consecutive intervals of $\mathbb N$ and denote by $N_p$ the cardinality of the interval $I_p$. Put $\alpha_k=0$ if $k\in I_{p}$ for some odd $p$, and $\alpha_k=\frac{2\pi}{N_p}$ if $k\in I_p$ for some even $p$. Then $e^{iA_k}=1$ if $k\in I_p$ for some odd $p$, and the $e^{iA_k}$, $k\in I_p$ enumerate the $N_p$ roots of $1$ is $p$ is even (starting with $e^{i{2\pi}/{N_p}}$ and ending with $1$). Hence $$\sum_{k\in I_p} e^{iA_k}=\left\{ \begin{matrix} N_p&p\;{\rm odd}\cr 0&p\; {\rm even} \end{matrix}\right.$$ It follows that if the sequence $(N_p)$ is very fast increasing (say $N_1+\dots N_p=o(N_{p+1})$) and if we put $n_p=N_1+\dots +N_p$, then $\frac{1}{n_{2l}}\sum_{k=1}^{n_{2l}} e^{iA_k}\to 0$ as $l\to\infty$, whereas $\frac{1}{n_{2l+1}}\sum_{k=1}^{n_{2l+1}} e^{iA_k}\to 1$.
• I was trying to figure out: $a_k$ sequence of real numbers tending to 0. Then in your example it seems to me, that $a_k$ can't be tending to 0. Can you explain to me this moment? Sorry if this is clear.
• The sequence $(\alpha_k)$ does tend to $0$: if $k$ is large, the index $p=p(k)$ such that $\alpha_k\in I_p$ is large too, so $N_{p(k)}\to\infty$ as $k\to\infty$. Jun 11 '13 at 10:39