Exercise 21, § 5

Let $x$ denote the identity function of $\mathbf{R}$. Let $\mathscr{M}_m(\mathbf{R})$ denote the set of positive measures $\nu$ on $\mathbf{R}$ such that $x^n\in \mathscr{L}^1(\mathbf{R}, \nu )$ for every integer $n\in \mathbf{N}$. a) Let $\nu \in \mathscr{M}_m(\mathbf{R})$ and put $\mu_n=\nu (x^n)$ for every $n\in \mathbf{N}$ ("moments of the measure $\nu$"). For every integer $n\geqslant 0$ and for every complex numbers $a_0, . . . , a_n$, one has

In what follows, we propose to study the converse assertion to question a) ("Hamburger moment problem"). In the rest of this exercise, we therefore suppose given a sequence $\boldsymbol{\mu}= (\mu_n)_{n\geqslant 0}$ of real numbers satisfying condition (24). b) There exists a unique positive sesquilinear form $b_{\boldsymbol{\mu}}$ on $\mathbf{C}[X]$ such that $b_{\boldsymbol{\mu}}(X^i,X^j) =\mu_{i+j}$ for all integers $i$ and $j$.

Let $E_{\boldsymbol{\mu}}$ denote the separated completion Hilbert space of $\mathbf{C}[X]$ for the form $b_{\boldsymbol{\mu}}$. c) The linear mapping $u_{\boldsymbol{\mu}}$ of $\mathbf{C}[X]$ into $\mathbf{C}[X]$ defined by $u_{\boldsymbol{\mu}}(P) =$ XP induces, by passing to the quotient, a symmetric partial operator $u_{\boldsymbol{\mu}}$ on the Hilbert space $E_{\boldsymbol{\mu}}$.

d) The partial operator $u_{\boldsymbol{\mu}}$ admits a self-adjoint extension.

e) There exists a measure $\nu \in \mathscr{M}_m(\mathbf{R})$ such that $\mu_n=\nu (x^n)$ for every $n\in \mathbf{N}$. f) The sesquilinear form $b_{\boldsymbol{\mu}}$ is separated if and only if, for every measure $\nu \in \mathscr{M}_m(\mathbf{R})$ such that $\mu_n=\nu (x^n)$ for every $n\in \mathbf{N}$, the support of $\nu$ is infinite. (Observe that the left-hand side of inequality (24) is equal to $b_\mu(p, p)$ for $p=a_0+\cdots +a_nX^n.$)