Integral kernel of Hilbert-Schmidt operator on $L^2(\mathbb{R})$

Question

Every compact operator $A: L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$ can be expressed in canonical form as $$ A=\sum_n p_n \left<\phi_n,\cdot\right> \psi_n, $$ where $\{\phi_n\}$ and $\{\psi_n\}$ are (not necessarily complete) orthonormal sets and $\{p_n\}$ is a sequence of positive numbers with $p_n\rightarrow 0$. If additionally $\{p_n\}$ is square summable, then $A$ is Hilbert-Schmidt operator. We know that every Hilbert-Schmidt operator on $L^2(\mathbb{R})$ has integral kernel $K(x,y)\in L^2(\mathbb{R}^2)$, but is it true in general that $$ K_A(x,y)=\sum_n p_n \psi_n(x)\overline{\phi}_n(y) $$ is the integral kernel of operator $A$? Do you know any books or papers touching upon this problem?