Obtaining the integral kernel of an operator

There is one-to-one correspondence between $L^2$ integral kernels and Hilbert-Schmidt operators on $L^2$. If $A$ is HS on $L^2$, and $(e_j)$ is an orthonormal basis for $L^2$, then the kernel can be expressed as $$K(x,y)=\sum\limits_{j,k}\langle Ae_j,e_k\rangle \overline{e_j}e_k.$$

Here is one especially convenient scenario: If the operator is pseudodifferential, and you know the symbol $a$, then you can alternatively express the kernel in terms of an oscillatory integral involving the symbol (which is, of course, equivalent to the above):

$$K(x,y)=(2\pi)^{-n}\int\limits_{\mathbb{R}^n} a(x,\xi) e^{i(x-y)\cdot\xi}\, d\xi. $$ I'd say that I personally look at it in the latter sense more frequently, though this likely depends on the work that you do.