Let $(X,\mu)$ be a $\sigma$-finite measure space. If $K\in\mathcal{L}^2(X\times X,\mu\times\mu)$ then the map $A_K:\mathcal{L}^2(X,\mu)\to\mathcal{L}^2(X,\mu)$ defined by\begin{equation} A_Kf(x)=\int_XK(x,y)f(y)d\mu(y) \end{equation} is Hilbert-Schmidt.
But Arveson (Proposition 2.8.6) says this $K\mapsto A_K$ is an isomorphism from $\mathcal{L}^2(X\times X,\mu\times\mu)$ to the space of Hilbert-Schmidt operators on $\mathcal{L}^2(X,\mu)$.
So in particular this map is onto. I do not know how to prove this. I tried to focus on the easiest case $X=[0,1]$ but still got no progress.
Can someone give a hint? Thanks!