This is purely a question sprung out of curiosity. If I have some $K(x,y)\in L^2(\mathbb{C}^2)$ and define the corresponding integral operator on $L^2(\mathbb{C})$
$$ L_K(f)=\int f(y)K(x,y)dy $$
is there anything interesting that happens when we act this operator on its own kernel with both arguments identified? ie, if I define $K(x):=K(x,x)$, is there anything interesting about the action of $L_{K(x,y)}$ on $K(x)$?
If $K$ is in $L^2({\mathbb C})$ then $K$ is actually not a function, but rather an equivalence class of square integrable functions modulo equality a.e. Since the diagonal $$ \Delta :=\{(x,x): x\in {\mathbb C}\} $$ has measure zero, you may change the values of $K$ randomly on $\Delta $, without changing the class of $K$ and, most importantly, without changing the integral operator.
I'd therefore say that your question is not a well posed question, as $K(x,x)$ can be anything you want.