I am getting acquainted with the beautiful theorem known as Jacobs–de Leeuw–Glicksberg decomposition. A special case of this theorem is the following:
Theorem. (Jacobs–Glicksberg–de Leeuw decomposition). Let $\mathcal{H}$ be a Hilbert space and let $T \in \mathcal{L}(\mathcal{H})$ have relatively weakly compact orbits. Then we have decomposition $\mathcal{H} = \mathcal{H}_r \oplus \mathcal{H}_r$, where : $$ \mathcal{H}_r = \overline{\operatorname{span}} \{x \in \mathcal{H} \ : \ \exists \alpha \in \mathbb{C}, |\alpha| = 1, Tx = \alpha x \} $$ $$ \mathcal{H}_s = \{ x \in \mathcal{H} \ : \ 0 \text{ is weak accumulation point of } \{T^n x \}_n \}$$
I am trying to understand the full strength of this statement. In particular, I would like to understand what $\mathcal{H}_s$ looks like. Embarrassingly, I can't seem to find an operator for which this decomposition would be non-trivial and yet understandable.
I would in particular very much like to see what an situation where $\mathcal{H}_s$ contains $x$ such that $T^n x $ does not have a limit in the weak topology (merely accumulation points). The weakly compact semigroup generated by $T$ will contain a projection onto $\mathcal{H}_r$, which follow from a more abstract version of the theorem. It seems it should normally contain also other projections, but I would like to see a case when this actually happens. (I think these two questions are strongly related). It would be absolutely perfect if $T$ happened to be a Koopman operator for a dynamical system, but all examples are appreciated.
Thank you in advance for your help.