In a lecture note, in the demonstration of a result it is said that "using analytic perturbation theory" it is possible to deduce certain things. The reference is Kato's classic book, "Perturbation theory for linear operators". I'm not very familiar with the book and couldn't find the exact result or collection of results that would allow me to reach the conclusion made.
The result is very intuitive, a small perturbation in the operator leads to a small perturbation in the spectrum.
Let $X$ a Banach space (possibly infinite dimensional) and $R: z \mapsto \text{Hom}(X,X)$ where the spectrum of $R(1)$ consists of an isolated simple eigenvalue at $1$ and a compact subset of $\mathbb{D}$.
From this i know that the spectrum of $R(1)$ consists of a subset of $\{ \lambda\in \mathbb{D}: |\lambda|\le \tau \}$ for some $\tau \in (0,1)$ and a simple isolated eigenvalue at $1$ with eigenprojection $P$. Fix some $\kappa\in(\tau,1)$ and let $R_A$ be a perturbation of $R(1)$, more precisely, $R_A(z)=R(1)+(z-1)R'(1)$, (assume that R'(1) is well defined.)
We have that $z\mapsto R_A(z)$ is holomorphic, so, here it is mentioned that Kato's book (analytic perturbation theory) is used, to deduce that:
Exists $\delta$ and $P_A : B_{\delta}(1) \to \text{Hom}(X,X)$, $\lambda_A :B_{\delta}(1) \to \mathbb{C}$ holomorphic such taht:
- $\lambda_A(z)$ is a simple eigenfunction of $R_A(z)$ with eigenprojection $P_A(z)$;
- $|\lambda_A (z)|> \kappa$ and the rest of the spectrum of $R_A(z)$ is contained in $\{ \lambda \in \mathbb{C} : |\lambda|\le\kappa \}$
As I said, I could not find this result in the book and I would like to know from someone who is familiar with the theory of perturbation, any reference to this result or the collection of result that allows to conclude what was said.
Edit: I believe that one of the results used may be Theorem 3.16 from Chapter 4 of Kato's book. The theorem can be seen here.