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.


1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.