Hankel operators and the Stieltjes moment problem
Let \(s\) be a non-vanishing Stieltjes moment sequence and let \(\mu\) be a representing measure of it. By \(\mu_n\) denote the image measure in \({\mathbb C}^n\) of \(\mu\otimes\sigma_n\) under the map \((t,\xi)\mapsto\sqrt{t}\xi\), where \(\sigma_n\) is the rotation-invariant probability measure on the unit sphere. By \({\mathcal A}(s)\) denote the space of holomorphic functions \(f(z)=\sum_{\alpha\in{\mathbb N}_0^n}a_\alpha z^\alpha\) satisfying \(\sum_{\alpha\in{\mathbb N}_ 0^n}{\alpha! s_{|\alpha|}\over (|\alpha|+n-1)!}|a_\alpha|^2<+\infty\). It is shown that this set is a reproducing kernel Hilbert space. On this space, consider the Hankel operator \(H_f\) with symbol \(f\). For a polynomial \(f\), let \(\text{Spec}(f)\) be the set of multi-indices \(k\in{\mathbb N}_ 0^n\) such that \({\partial^k f\over\partial z^k}(0)\neq 0\). One of the main results of the paper says that, if \(f\) is a holomorphic polynomial, then {\parindent=6mm \begin{itemize}\item[(a)] \(H_{\overline{f}}\) is bounded if and only if \[ \sup_{d\in {\mathbb N}_0}\left({s_{d+2|k|}\over s_{d+|k|}}-{s_{d+|k|}\over s_d}+{n-1\over d}{s_{d+|k|}\over s_d}\right)<+\infty \] for \(k\in\text{Spec}(f)\); \item[(b)] \(H_{\overline{f}}\) is compact if and only if \[ \lim_{d\to +\infty}\left({s_{d+2|k|}\over s_{d+|k|}}-{s_{d+|k|}\over s_d}+{n-1\over d}{s_{d+|k|}\over s_d}\right)=0 \] for \(k\in\text{Spec}(f)\); \item[(c)] If \(p>0\), then \(H_{\overline{f}}\) belongs to the Schatten class \(S_p\) if and only if \[ \sum_{d\in{\mathbb N}}d^{n-1}\left({s_{d+2|k|}\over s_{d+|k|}}-{s_{d+|k|}\over s_d}\right)^{p/2}+(n-1)d^{n-1-p/2}\left({s_{d+|k|}\over s_d}\right)^{p/2}<+\infty \] for all \(k\in\text{Spec}(f)\). \end{itemize}} In particular, it is shown that, if \(n=1\), then there are nontrivial Hilbert-Schmidt Hankel operators with anti-holomorphic symbols if and only if \(s\) is exponentially bounded. In this case, the space of symbols of such operators is the classical Dirichlet space.
scientific article; zbMATH DE number 5655705
scientific article