Unbounded Helson Forms with a Moment Representation

We study the positive Helson form $h_0[a,b]={\sum }_{m,n\ge 1}\alpha (mn)a_n\overline{b_m}$, $a,b\in c_{00}$, where $c_{00}$ denotes the finitely supported sequences in ${\mathrm{\ell }}^2(\mathbb{N})$, and $\alpha (n)={\int }_{\mathbb{R}}{n}^{-s}d\mu (s)<\mathrm{\infty }$ for a finite positive Borel measure $\mu$. Under the Carleman condition ${\sum }_{k\ge 1}\alpha (4^k{)}^{-1\mathrm{/}(2k)}=\mathrm{\infty }$, closability of $h_0$ is equivalent to $\sqrt{n}\alpha (n)\to 0$, and hence to $\mu ((-\mathrm{\infty },1\mathrm{/}2])=0$. For measures concentrated on $(1\mathrm{/}2,\mathrm{\infty })$, let $N_0:{\mathrm{\ell }}^2(\mathbb{N})\supset c_{00}\to L^2(\mu )$ be given by $(N_{0}a)(s)={\sum }_{n\ge 1}{a}_n{n}^{-s}$. We prove that $N:=\overline{N_0}=N_{\max }$, where $N_{\max }$ acts by the same Dirichlet series on the domain $\mathrm{Dom}(N_{\max })=\{a\in {\mathrm{\ell }}^2(\mathbb{N}):{\int }_{1\mathrm{/}2}^{\mathrm{\infty }}\mathrm{\mid }{\sum }_{n\ge 1}{a}_n{n}^{-s}{\mathrm{\mid }}^{2}d\mu (s)<\mathrm{\infty }\}$. The closure $h$ of $h_0$ consequently satisfies $\mathrm{Dom}(h)=\mathrm{Dom}(N)$ and $h[a,b]=⟨Na,Nb{⟩}_{L^2(\mu )}$, and its associated positive self-adjoint operator is $H_{\mu }=N^{\ast }N$. We further prove that the sequences $a\in c_{00}$ satisfying ${\sum }_{n\ge 1}{a}_n\mathrm{/}\sqrt{n}=0$ form an operator core for $H_{\mu }$. The bounded positive Helson matrices ${\mathcal{H}}_{{\mu }_t}$ generated by $d{\mu }_t(s)=(1-e^{-t(s-1\mathrm{/}2)})d\mu (s)$ give a characterization of its operator domain: $g\in \mathrm{Dom}(H_{\mu })$ if and only if ${\mathcal{H}}_{{\mu }_t}g$ converges in ${\mathrm{\ell }}^2$ as $t\to \mathrm{\infty }$, with limit $H_{\mu }g$. Moreover, $H_{\mu }$ is trace class if and only if ${\sum }_{n\ge 1}\alpha (n^2)<\mathrm{\infty }$, and for $\alpha \in {\mathrm{\ell }}^2$ it agrees with the maximal operator defined by the matrix $(\alpha (mn){)}_{m,n\ge 1}$.