Given a measure space $(X, \mu)$ and a measurable integral kernel $k : X \times X \rightarrow \mathbb{C}$, the operator 
$$ K f(\xi) =\int\_{X} f(x) k(x,\xi) d \mu(x),$$
the operator $K$ is Hilbert Schmidt iff $k \in L^2(X \times X, \mu \otimes\mu)$! 


Q1:The main point of this questions, what are necessary and sufficient conditions for it to be trace class? 


I know various instances, where
$$ \mathrm{tr} K = \int\_X k(x,x) d \mu(x).$$


Q2:What are counterexamples, where $x \mapsto k(x,x)$ is integrable, but the operator is not trace class?


Q3:What are counterexamples for a $\sigma$ finite measure space, where $k$ is compactly supported and continuous, but the kernel transformation is not trace class and the above formula fails?


Q4: Is there a good survey/reference for these questions.



When is an integral transform trace class?
There are many results of the kind you ask about in the book


I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators. Providence, RI: American Mathematical Society, 1969.


It contains both necessary and sufficient conditions, and counter-examples.


