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This is a problem from my exam on functional analysis. I did only trivial case and now I'm just curious about another cases.

Let $T : H \rightarrow H$ is a linear continuous unitary ($T^*=T^{-1}$) operator, $H$ is a Hilbert space (not necessary). $T_n$ $-$ a sequence of linear operators $T_n \underset{n\rightarrow \infty}{\longrightarrow} T$, $|| T_n - T||\rightarrow 0$.

Suppose that for some subspace $H_1 \subset H$, $\forall h_1 \in H_1 \Rightarrow Th_1=h_1$ and we have a sum on it where $n \rightarrow \infty$ :

$$ S_n= \frac{1}{n}\left( T_1 h_1 + T_1 T_2 h_1 + \dots + T_1 \dots T_n h_1 \right), \ \ h_1 \in H_1. $$

The question was:

Find such sufficient сonditions on $T_k$ and $\ T$ that (Limit exist):

$$ \underset{n\rightarrow \infty} \lim S_n = h^* \in H $$

I could construct only trivial example when for all $T_k$ we have $T_k h_1 = h_1$.

Conditions $|| T_k|| < 1$ or $||T_1 T_2 ... T_k || < 1$ not sufficient. My question is about examples of some others sufficient conditions (may be even necessary and sufficient)?

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