Suppose $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$. Consider the Newton potential \begin{equation} T [\phi](x) = \int_{\Omega} \frac{1}{|x-y|} \phi(y)dy. \end{equation} It is well know that $T$ is a bounded linear operator from $L^2(\Omega)$ to $H^2(\Omega)$. Hence it is a self adjoint compact operator defined on $L^2(\Omega)$. Suppose that it has the following spectral decomposition: $$T \phi = \sum^\infty_{j = 1}\lambda_j (\phi,e_j) e_j,$$ where $(\lambda_j,\phi_j)$ is the eigenpair counting multiplicity. And we can see $ker T = \{0\}$ from the following observation: $\Delta T[\phi] = C\phi$ on $\Omega$ for some positive constant $C$.

We say that a vector $q$ in $L^2(\Omega)$ is a quasinilpotent vector if $$ \lim_{n \to \infty}||T^n q||^{\frac{1}{n}} = 0. $$ Then from above spectral decomposition and fact that $\lambda_j > 0$, we can claim all the quasinilpotent vectors of $T$ vanish. Indeed, if $\phi$ is a quasinilpotent vector, then $$ \lim_{n \to \infty}|(e_j,T^n \phi)|^{\frac{1}{n}} = \lambda_j |(e_j,\phi)|^{1/n} = 0 ,$$ which gives us $(e_j,\phi)$ vanishes for all $j$.

I would like to prove the same result (all the quasinilpotent vectors vanish) for the following operator, $$ T_k[\phi] = \int_{\Omega} \frac{e^{ik|x-y|}}{|x-y|}\phi(y)dy,$$ which is a also a compact operator on $L^2(\Omega)$. But we may not expect the above arguments work in our case since the spectral structure of $T_k$ is not clear. Perhaps we need turn to elliptic PDE theory for help.

Thank you very much in advance for any insight or suggestions.