Define an integral operator $T$ on $L^2([0,1])$ by $$Tf(x) = \int\int K(x, y) f(y) \, dx\, dy$$. Such an operator is Hilbert-Schmidt when $K$ is in $L^2([0,1]\times [0,1])$.

I heard that if $K$ is smooth, then $T$ is in fact trace-class. Why is this?

When is such an operator trace-class?

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