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?