Unbounded Helson Forms with a Moment Representation

Abstract

We study the positive Helson form $h_0[a,b]=\sum_{m,n\ge1}\alpha(mn)a_n\overline{b_m}$ on the finitely supported sequences $c_{00}\subset\ell^2(\mathbb{N})$, where $\alpha(n)=\int_{\mathbb{R}}n^{-s}d\mu(s)<\infty$ for a finite positive Borel measure $\mu$. Under the Carleman condition $\sum_{k\ge1}\alpha(4^k)^{-1/(2k)}=\infty$, closability of $h_0$ is equivalent to $\sqrt{n}\alpha(n)\to0$, and hence to $\mu((-\infty,1/2])=0$. For measures concentrated on $(1/2,\infty)$, we identify the closure of the operator $N_0a=\sum_{n\ge1}a_nn^{-s}$, defined on $c_{00}$, with its maximal realization from $\ell^2(\mathbb{N})$ to $L^2(\mu)$. We further prove that the sequences $a\in c_{00}$ satisfying $\sum_{n\ge1}a_n/\sqrt{n}=0$ form an operator core for the positive self-adjoint operator associated with the closure of $h_0$.