On this week I'm trying to show the subspace $\mathcal{N (A)} \bigoplus \overline{Im(A)}$ is closed and the equation
$$ \lim_{t \to \infty} \frac{1}{t} \int_b^t T(s)u ds = \lim_{t \to \infty} \frac{1}{t} \int_0^t T(s)uds$$
holds for $u \in \mathcal{N (A)} \bigoplus \overline{Im(A)}$ and $b \geq 0$.
The book gives the following steps to achieve this result:
Let $X$ be a Banach Space, {$T(t) : t \geq 0$} $\subset \mathcal{L}(X)$ a contraction $C_0$-semigroup and $A:D(A) \subset X \to X$ your infinitesimal generator. Then,
(1) For every $u \in \mathcal{N}$ and $v \in \overline{Im(A)}$,
$$ \lim_{t\to\infty} \frac{1}{t} \int_0^t T(s)(u+v)ds = u.$$
(Note that $T(t)u = u, t \geq 0$)
(2) The subspace $F$ of points $u$ such the limit
$$ \lim_{t\to\infty} \frac{1}{t} \int_0^t T(s)u ds \quad (:= Pu) $$
exists is closed.
(3) Show that $P \in \mathcal{L}(F,X)$, $T(t)F \subset F$ for every $t \geq 0$, $PT(t)u = Pu$ for every $u \in F$ and $P^2u = Pu$ for every $u \in F$.
(4) Conclude $P$ is a projection of $F$ into $\mathcal{N}(A)$ where $P(\overline{Im(A)}) = {0}$.
Attempts:
For step (1), I noticed the problem could be solved if I prove
$$ \lim_{t\to\infty} \frac{1}{t} \int_0^t T(s)v ds = 0. $$
Then, I tried to use $\frac{d}{ds}T(s) = T(s)A$ and the formula
$$ T(t)u = \lim_{N\to\infty} \frac{1}{2\pi i} \int_{1-iN}^{1+iN} e^{\lambda t} (\lambda-A)^{-1} u d\lambda $$
with a sequence $Au_n \to v$ but didn't get any far.
For step (2) I am following the same way, but my intuition is telling me it's not here, I guess it has to directly use step (1).
In the steps (3) and (4) I didn't get any idea at all. Every kind of help would be very appreciated!