I study integral operators and I have several questions for which I didn't find answers. I'm looking for references in the first place, books, where this questions discussed. Here are the questions:
- When kernel determines an operator? It's well known, that for any measure space $(X,\mu)$ and $k\in L^2(X\times X)$ operator $$K:L^2(X)\rightarrow L^2(X)\ \ \ \ \ (Kf)(x)=\int\limits_X k(x,y) f(y)\mu(dy)$$ is well-defined, compact and $\|K\|\le \|k\|_{L^2}$. My question is: are there more general statements on when some function $k(x,y)$ defines such an operator in $L^2$?
- When integral operator is self-adjoint? It's clear that if $k(x,y)=\overline{k(y,x)}$, thet $K$ is self-adjoint. Is the converse also true?