Chains of generalised eigenvectors of an operator pencil

Question

I am reading a paper "Finding Eigenvalues of Holomorphic Fredholm Operator Pencils using boundary value problems and Contour integrals" of Beyn, Latushkin, Rottmann-Matthes and in subsection 2.3 they mention the definition of a chain of generalised eigenvectors of an operator pencil:

Let $\Omega\subseteq \mathbb{C}$ be some open domain and $X,Y$ be Banach spaces. Let $\mathbb{H}^0(X,Y)$ denote the space of Fredholm operators of index $0$ from $X$ to $Y$.

Definition 1. Let $F: \Omega\rightarrow \mathbb{H}^0(X,Y)$ be a holomorphic Fredholm operator pencil and $\lambda_0\in\sigma(F)$. A tuple $(v_0,...,v_{n-1})\in X^n$ is called a chain of generalised eigenvectors (CGE) if the polynomial $v(\lambda)=\sum_{j=0}^{n-1}(\lambda-\lambda_0)^j v_j$ satisfies $$ (Fv)^{(j)}(\lambda_0)=0,j=0,..,n-1. $$

Definition 2. In Mennicken and Möller, "Non-self-adjoint Boundary Eigenvalue Problems" they say that $(v_0,..,v_{n-1})\in X^n$ is a chain of an eigenvector and associated vectors (CEAV) if $v$ (defined the same as above) is a root function of $F$ at $\lambda_0$, that is $v(\lambda_0)\neq 0$ and $(Fv)(\lambda_0)=0$, and additionally require the order of the zero of $Fv$ at $\lambda_0$ to be larger or equal $n$.

Questions.

1. The Definitions should be equivalent (modulo the fact that they call the same object slightly different), however in Definition 2 we immediately get $v_0\neq 0$ whereas in Definition 1 we don't have such a requirement. How do we see that both definitions are indeed equivalent?

2. In both Definitions I don't understand the expression $(Fv)(\lambda_0)$. How can we compose a function $F: \Omega\rightarrow \mathbb{H}^0(X,Y)$ with a function $v:\Omega\rightarrow X$? If we simply evaluate both and then compose: $$ (Fv)(\lambda_0)=F(\lambda_0)v(\lambda_0)\in Y, $$ we run in the problem that $v(\lambda_0)=v_0$ and hence this would ruin the definitions. So we somehow have to let $F$ and $v$ be unevaluated and compose them, but how? Can somebody give me an example so that I understand how this works?

3. Whatever $(Fv)(\lambda_0)$ can mean, I am sure that it's value should be in $Y$. But then we run into another problem in Definition 1, how can we compose it multiple times $(Fv)^{(j)}(\lambda_0)$?

Remark. If you plug in $X=Y=\mathbb{R}^d$ and take $F(\lambda)=A-\lambda I$ then we should probably get that a CGE (or CEAV) $(v_0,...,v_{n-1})$ of $F$ at some $\lambda_0$ should consist out of generalised eigenvectors $v_j$ of $A$.