Let $X$ be a Banach space and $T \in \mathcal L(X)$.

The authors Engel and Nagel introduce in their book "One-Parameter Semigroups for Linear Evolution Equations" on p. 248 the concept of the **Fredholm domain** of $T$ defined by
$$\rho_F(T) := \{\lambda \in \mathbb C: \lambda - T \text{ is a Fredholm operator} \}.$$
On the next page the following is stated:

*"Here, we only recall that the poles of $R(\cdot, T)$ with finite algebraic multiplicity belong to $\rho_F(T)$. Conversely, an element of the unbounded connected component of $\rho_F(T)$ either belongs to $\rho(T)$ or is a pole of finite algebraic multiplicity."*

I can prove the first statement very elementary just by using properties of spectral projections and some very basic functional calculus. But the second statement seems to be quite difficult to prove. In the cited literature I found a proof of the stament (cf. the proof Corollary XI.8.5 in "Classes of Linear Operators Vol. I" by Gohberg, Goldberg and Kaashoek). But it seems to rely on quite some theorems about Fredholm operator valued functions.

So my question is whether there is a more elementary way to see that the statement holds, maybe just by using some basic facts on spectral projections? I thought quite some time about it but couldn't prove it. So is there maybe a reference for the statement which uses more elementary arguments? Or does someone know another way how to prove it? I am looking forward to your answers.