This is a more specific version of a question I asked before without much luck. I believe this should be standard perturbation theory, but looking at Kato's book has not helped. Any references would be appreciated!

Let $X_n$ be a Markov chain on a compact metric space $(\Omega,d)$ with generator $\mathcal{L} \colon \mathcal{B}(\Omega) \to \mathcal{B}(\Omega)$, where $\mathcal{B}(\Omega)$ is the Banach space

$\mathcal{B}(\Omega) = \{ f \colon \Omega \to \mathbb{R} : \|f\|_{Lip}+ \|f\|_\infty < \infty\}$. (It doesn't really matter what the Banach space is)

Suppose the chain has a unique invariant measure $\mathbb{Q}$. Consider a perturbed Markov chain $X_n^\epsilon$ with Markov operator $\mathcal{L}_\epsilon = \mathcal{L} + \epsilon \Delta$, where $\Delta({1}) \equiv 0$ and $\|{\Delta}\| \leq 1$.

Suppose that for $\epsilon$ small enough, $\mathcal{L}_\epsilon$ has a unique invariant measure $\mathbb{Q}_\epsilon$.

Given that $\mathcal{L}_\epsilon$ is a compact operator from $\mathcal{B}(\Omega)$ to itself for $\epsilon$ small enough, with spectral gap independent of $\epsilon$, then for any test function $g\in\mathcal{B}(\Omega)$, is

$\epsilon \mapsto \mathbb{Q}_\epsilon(g)$

a real analytic function of $\epsilon$ around $0$?

Recognizing that the invariant measure is an eigenfunction of the adjoint operator has not really led to anything fruitful, though I'm by no means an expert in the field. Any help would be much appreciated!