Self-adjoint operators associated with Hankel moment matrices

In a paper from 2016 D. R. Yafaev initiated a study of closable Hankel forms associated with the moments $(m_n)$ of a positive measure with infinite support on the real line. If the measure is concentrated on the open interval $(-1,1)$, Yafaev characterized the closure of the form, but he left aside a precise description of the correponding positive self-adjoint Hankel operator in the Hilbert space of square summable sequences. We discuss the self-adjoint Hankel operators associated with closed Hankel forms, not only in the case studied by Yafaev, but also in two other cases, where the Hankel form is closable, namely if the moment sequence is indeterminate or if the moment sequence is determinate with finite index of determinacy. We recall in passing the result of Yafaev that the Hankel form is not closable if the moment sequence is determinate in the strong sense, say if it satisfies a Carleman condition.