Let $\mathcal{H}$ be a Hilbert space. Let $\{U(t)\}_{t \in \mathbb{R}}$ be a strongly continuous unitary group, such that $\forall f \in \mathcal{H}$:
$$f(t + \theta) = U(t)f(\theta)U(t)^{-1},\; \theta \in \Omega,$$
where $\Omega := \{z \in \mathbb{C}: 0 < \operatorname{Im}(z) < a\}, \;a \in \mathbb{R}$ fixed.
Let $A$ be the generator of $\{U(t)\}_{t \in \mathbb{R}}$, that is: $U(t) = \exp(-itA)$. $A$ is an unbounded operator on $\mathcal{H}$. Let $P(\cdot)$ be the spectral projection for $A$. Let $D:= \{u \in \mathcal{H}: P([-M,M])u = u\},$ for some fixed $M.$ I have the following statements, which I can't justify:
(1) $D$ is dense in $\mathcal{H}.$
(2) $ \exp(-iwA)D = D, \;\forall w \in \mathbb{C}.$
Could someone help me to understand please ?
If $D$ is defined as you wrote (with a fix $M$) the it is not dense unless $A$ is bounded.
The density of $D=\{u\in H: \ \exists M\ \text{ such that } P([-M,M])u=u\}$ follows from the fact that $P(\mathbb R)=I$ and $ P([-M,M])\to I$ in the strong operator topology.
The second equality is due to the fact that $P([-M,M])$ commutes with $\exp(-iwA)$ via Borel Functional Calculus if $P([-M,M])u=u$, then $$ P([-M,M])\exp(-iwA)u=\exp(-iwA)P([-M,M])u=u=\exp(-iwA)u. $$