Suppose $k:\mathcal{X} \times \mathcal{X} \to \mathbb{R}$ be a symmetric positive (semi-)definite kernel. The Moore-Aronszajn Theorem indicates that there is a reproducing kernel Hilbert Space $\mathcal{H}$ associated with $k$. Let $\mu$ be a distribution on $\mathcal{X}$ and consider the integral operator $S_k(f): L^2(\mu) \to \mathcal{H}$ for all $f\in L^2(\mu)$, defined as $S_k(f) = \int_\mathcal{X} k(\cdot,x')f(x') d\mu(x')$. The spectral decomposition theorem ensures that $S_k$ admits an eigenvalue decomposition. Let $(\lambda_k,\phi_k)_k$ represent the eigenvalue-eigenfunction pair of $S_k$.
Now, I define a centered kernel $g(x,x') = k(x,x') -\int_\mathcal{X}k(x,y)d\mu(y) -\int_\mathcal{X}k(y,x')d\mu(y) + \int_{\mathcal{X}\times \mathcal{X}}k(y,z)d\mu(y) d\mu(z)$ and consider the integral operator $S_g$ defined with respect to $g$. Can one establish a relationship between the eigenvalues, say $\tilde{\lambda}_i$, of $S_g$ and $\lambda_i$ ? In particular, can we show if $\lambda_i$ decay as $i^{-a}$ for some $a>0$, $\tilde{\lambda}_i$ also decays as $i^{-b(a)}$ for some function $b$ ?
