Perturbation of eigenvalues and eigenfunctions of a linear operator

Question

I came across this Wikipedia page giving approximate formulas for eigenvalues and eigenvectors of a perturbed matrix. Namely,

Let $A' \in \mathbb{R}^{n \times n}$ be a real symmetric positive definite matrix with distinct eigenvalues $\{t_i\}_{i=1}^n$ and their corresponding orthonormal eigenvectors $\{v_i\}_{i=1}^n$. Assume that $t_1 > t_2 > \cdots > t_n$. Let $E \in \mathbb{R}^{n \times n}$ a real symmetric matrix. Consider a perturbation $A$ of $A'$ given by $A=A'+E$, with the eigenpairs of $A$ denoted by $\{(s_i,w_i)\}_{i=1}^n$. The approximated eigenvectors of $A$ are given by

$$ \widetilde{w}_{i} = v_{i} + \sum_{k=1, k \neq i}^{n} \frac{(Ev_i,v_k)}{t_i-t_k}v_k + O(\|E\|_2^2), \quad 1 \leq i \leq n , $$

and the approximated eigenvalues of $A$ are given by

$$\widetilde{s}_i = t_i + v_i^TEv_i + O(\|E\|_2^2), \quad 1 \leq i \leq n $$

My question is if there exists similar formula for the eigenvalues and eigenfunctions of a linear operator? i.e., to linear operators in an infinite dimensional linear space.

Answer 1

See the canonical reference:

Kato, Tosio, Perturbation theory for linear operators., Classics in Mathematics. Berlin: Springer-Verlag. xxi, 619 p. (1995). ZBL0836.47009.

Also see:

Albeverio, Sergio; Kuzhel, Sergei; Nizhnik, Leonid P., On the perturbation theory of self-adjoint operators, Tokyo J. Math. 31, No. 2, 273-292 (2008). ZBL1182.47013.