I'm hoping this is the correct place to post this. The context of this question is from physics but my question is purely mathematical. For some background, I am researching Von-Neumann Entropy in 1-dimensional boson models, and I have come across the following operator on $L^2(\mathbb{C}^2)$:
$$ L_G(f)=\int_A f(x_1,y)G(y,x_2)dy $$ where $A$ is some compact set and $G$ is some given element of $L^2(\mathbb{C}^2)$.
My question is as follows: Is this a well-studied operator? If so, where can I read about its properties? Any information on such operators would be highly useful. Thanks.