Let $\Omega \subset \mathbb R^2$ be a bounded domain with a smooth boundary. Let $$\bar{\partial}= \frac{1}{2} ( \partial_{x^1} + i \,\partial_{x^2}),$$ and let $G: L^2(\Omega)\to H^2(\Omega)$ be the Dirichlet Green's function on $\Omega$ that is defined by $GF:=u$ where $u\in H^2(\Omega)$ is the unique solution to $-\Delta u =F$ on $\Omega$ with $u|_{\partial \Omega}=0$.

Let us define the compact operator $T:L^2(\Omega)\to L^2(\Omega)$ via $$ T F = \bar{\partial}GF.$$

My question is to try to understand the spectrum of the operator $T$ and whether for instance it admits a discrete spectrum that provides a basis for $L^2(\Omega)$.