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 $n \rightarrow \infty$?

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

There are several cases:

  • 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 $T_j=\left(1+\frac 1{\log(j+1)}\right)I$).

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