# Prove that the limit exist

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. $T_n$ - 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 $n \rightarrow \infty$?

• Just to understand... your $T$ is the identity map, right? Jun 9 '13 at 7:55
• Yes, T is the identity map
– user59928
Jun 9 '13 at 8:04

• the limit exists and is $0$ (when $T_j=\left(1-\frac 1{j+1}\right)I$);
• the limit exists and is the identity (when $T_j=I)$;
• the limit doesn't exist (when $T_j=\left(1+\frac 1{\log(j+1)}\right)I$).