# fa.functional analysis – Compactness of integral operators

We know that if an operator has $$L^2$$-kernel, then it is Hilbert-Schmidt.
Is there a similar simple criterion to detect compact operators?

In particular, I’d like to know the following: Let $$f$$ be a Schwartz function on $${mathbb R}^2$$ with $$mathrm{supp}(f)subset{mathbb R}times J$$ for some compact Interval $$J$$.
Let
$$k(x,y)=f(e^x,x-y)$$
Is the operator $$T:L^2({mathbb R})to L^2({mathbb R})$$, $$T(phi)(x)=int_{mathbb R}k(x,y)phi(y),dy$$ a compact operator?