Exercise 22, § 5

We take up the notation of the preceding exercise again.

Let $\boldsymbol{\mu}= (\mu_n)_{n\in\mathbf{N}}$ be a sequence of real numbers satisfying conditions (24). We consider the question of the uniqueness of a measure $\nu \in \mathscr{M}_m(\mathbf{R})$ such that $\mu_n=\nu (x^n)$ for all $n\in \mathbf{N}$. We suppose that the sesquilinear form $b_{\boldsymbol{\mu}}$ is separating (cf. question f) of the preceding exercise).

a) If there exists a measure $\nu \in \mathscr{M}_m(\mathbf{R})$ with compact support such that $\nu (x^n) =\mu_n$ for all $n\in \mathbf{N}$, then $\nu$ is the unique measure in $\mathscr{M}_m(\mathbf{R})$ such that $\nu (x^n) =\mu_n$ for all $n\in \mathbf{N}$.

b) Let $\nu \in \mathscr{M}_m(\mathbf{R})$ be such that $\nu (x^n) =\mu_n$ for all $n\in \mathbf{N}$. For every $p\in \mathbf{C}[X]$, one has $p\in \mathscr{L}^2(\mathbf{R}, \nu )$ and $\|p\|=b_{\boldsymbol{\mu}}(p, p)$.

c) The canonical injection of $\mathbf{C}[X]$ into $\mathscr{L}^2(\mathbf{R}, \nu )$ defines by continuity and by passing to the quotient an isometry of $E_{\boldsymbol{\mu}}$ into $L^2(\mathbf{R}, \nu )$. In the sequel, we identify $E_{\boldsymbol{\mu}}$ with its image, which is a closed subspace of $L^2(\mathbf{R}, \nu )$.

d) One has $u_{\boldsymbol{\mu}}\subset m_x$, where $m_x$ denotes the operator of multiplication by $x$ in $L^2(\mathbf{R}, \nu )$.

e) The measure $\nu$ is the spectral measure of the constant function 1 relative to $m_x$.

f) Let $\nu \in \mathscr{M}_m(\mathbf{R})$. One has $\nu (x^n) =\mu_n$ for all $n\in \mathbf{N}$ if and only if there exist a Hilbert space F and a self-adjoint partial operator $v$ on F satisfying the following conditions:

(i) The space $E_{\boldsymbol{\mu}}$ is a closed subspace of F, and the canonical injection of $E_{\boldsymbol{\mu}}$ into F is an isometry;

(ii) One has $u_{\boldsymbol{\mu}}\subset v$;

(iii) The measure $\nu$ is the spectral measure of the constant function $1\in E_{\boldsymbol{\mu}}$ relative to $v$.

g) There exists a unique measure $\nu \in \mathscr{M}_m(\mathbf{R})$ such that $\nu (x^m) =\mu_n$ for all $n\in \mathbf{N}$ if and only if the partial operator $u_{\boldsymbol{\mu}}$ is essentially self-adjoint. h) Let $\nu_{lg}$ be the positive measure on $\mathbf{R}$ with support in $\mathbf{R}_+$ whose restriction to $\mathbf{R}_+^*$ has density

1 2

$t\mapsto$ exp($-$ log($t$)$/2$)

with respect to Lebesgue measure. Let $\widetilde{\nu}_{lg}$ be the positive measure on $\mathbf{R}$ with support in $\mathbf{R}_+$ whose restriction to $\mathbf{R}^*_+$ has density $t\mapsto 1 +$ sin(2$\pi$ log($t$)) with respect to $\nu_{lg}$.

We have $\nu_{lg}\in \mathscr{M}_m(\mathbf{R})$ and $\widetilde{\nu}_{lg}\in \mathscr{M}_m(\mathbf{R})$.

i) We have $\nu_{lg}(x^n) =\widetilde{\nu}_{lg}(x^n)$ for all $n\in \mathbf{N}$.