Unbounded Helson Forms with a Moment Representation

Abstract

We study the positive Helson form $h_0[a,b]=\sum_{m,n\ge1}α(mn)a_n\overline{b_m}$, $a,b\in c_{00}$, where $c_{00}$ denotes the finitely supported sequences in $\ell^2(\mathbb{N})$, and $α(n)=\int_{\mathbb{R}}n^{-s}dμ(s)<\infty$ for a finite positive Borel measure $μ$. Under the Carleman condition $\sum_{k\ge1}α(4^k)^{-1/(2k)}=\infty$, closability of $h_0$ is equivalent to $\sqrt nα(n)\to0$, and hence to $μ((-\infty,1/2])=0$. For measures concentrated on $(1/2,\infty)$, let $N_0:\ell^2(\mathbb{N})\supset c_{00}\to L^2(μ)$ be given by $(N_0a)(s)=\sum_{n\ge1}a_nn^{-s}$. We prove that $N:=\overline{N_0}=N_{\max}$, where $N_{\max}$ acts by the same Dirichlet series on the domain $\operatorname{Dom}(N_{\max})=\{a\in\ell^2(\mathbb{N}):\int_{1/2}^{\infty}|\sum_{n\ge1}a_nn^{-s}|^2dμ(s)<\infty\}$. The closure $h$ of $h_0$ consequently satisfies $\operatorname{Dom}(h)=\operatorname{Dom}(N)$ and $h[a,b]=\langle Na,Nb\rangle_{L^2(μ)}$, and its associated positive self-adjoint operator is $H_μ=N^*N$. We further prove that the sequences $a\in c_{00}$ satisfying $\sum_{n\ge1}a_n/\sqrt n=0$ form an operator core for $H_μ$. The bounded positive Helson matrices $\mathcal H_{μ_t}$ generated by $dμ_t(s)=(1-e^{-t(s-1/2)})dμ(s)$ give a characterization of its operator domain: $g\in\operatorname{Dom}(H_μ)$ if and only if $\mathcal H_{μ_t}g$ converges in $\ell^2$ as $t\to\infty$, with limit $H_μg$. Moreover, $H_μ$ is trace class if and only if $\sum_{n\ge1}α(n^2)<\infty$, and for $α\in\ell^2$ it agrees with the maximal operator defined by the matrix $(α(mn))_{m,n\ge1}$.