# Nuclearity properties of Integral operators with Schwartz type kernels

Consider an integral operator $$T:L^2({\mathbb R}^n)\to L^2({\mathbb R}^n),\qquad (Tf)(x)=\int_{\mathbb R^n} dy\,K(x,y)f(y),$$ with Schwartz type kernel $K\in{\mathscr S}({\mathbb R}^{2n})$ (smooth and rapidly decaying at infinity, for all derivatives). Then $T$ is Hilbert-Schmidt (in particular, compact) with Hilbert-Schmidt norm $\|K\|_2$, i.e. the singular values $\mu_k$ of $T$ satisfy $$\sum_k\mu_k^2=\int_{\mathbb R^n} dx\int_{\mathbb R^n}dy\,|K(x,y)|^2<\infty.$$

are square summable. I wonder what the best statement on the decay of the $\mu_k$ is in this situation (Maybe $\sum_k\mu_k^p<\infty$ for all $p>0$)? I suppose this must be known (possible together with bounds on $\sum_k\mu_k^p$), but who knows a reference?

Edit: Under the stronger assumption that $K$ is Schwartz class and there exists a compact set $C\subset{\mathbb R}^n$ such that $K$ has support in $C\times{\mathbb R}^n$ or ${\mathbb R}^n\times C$, the conjectured $\sum_k\mu_k^p<\infty$ for all $p>0$ can be shown by multiplying $T$ with suitable test functions which are constant on $C$, and suitable powers of $(1-\Delta)$. But for general Schwartz kernels $K$ this argument does not apply.

To answer my own question: If the kernel $K\in{\mathscr S}({\mathbb R}^{2n})$ is of Schwartz type, then the associated operator $T$ is $p$-nuclear for all $p>0$, i.e. its singular values satisfy $\sum_k\mu_k^p<\infty$ for all $p>0$, i.e. they decay faster than any inverse power of $k$.
One way of seeing this is as follows: Consider the decomposition of $K$ into the orthonormal basis $\phi_{\underline{\alpha}}\otimes\phi_{\underline{\beta}}:=\phi_{\alpha_1}\otimes...\otimes\phi_{\alpha_{n}}\otimes \phi_{\beta_1}\otimes...\otimes\phi_{\beta_{n}}$ of $L^2({\mathbb R}^{2n})$, where the $\phi_i\in L^2({\mathbb R})$ denote the Hermite functions. Because $K$ is Schwartz class, one has $K=\sum_{\underline{\alpha},\underline{\beta}}c_{\underline{\alpha},\underline{\beta}}\cdot\phi_{\underline{\alpha}}\otimes\phi_{\underline{\beta}}$ with rapidly decreasing coefficients $c_{\underline{\alpha},\underline{\beta}}$, i.e. for each $p>0$ one has $\sum_{\underline{\alpha},\underline{\beta}}\left|c_{\underline{\alpha},\underline{\beta}}\right|^p<\infty$. This statement can for example be found in Reed/Simon I Thm. V.13.
Hence the operator $T$ can be written as $T\xi=\sum_{\underline{\alpha},\underline{\beta}}c_{\underline{\alpha},\underline{\beta}}\langle\phi_{\underline{\alpha}},\xi\rangle\,\phi_{\underline{\beta}}$, so that $\langle\phi_{\underline{\alpha}},T\phi_{\underline{\beta}}\rangle=c_{\underline{\beta},\underline{\alpha}}$. But the singular values satisfy $\sum_k\mu_k^p\leq\sum_{\underline{\alpha},\underline{\beta}}\left|\langle\phi_{\underline{\alpha}},T\phi_{\underline{\beta}}\rangle\right|^p$, which is finite by the previous argument.