Exercise 27, § 5

Let $M = X^2Y^2(X^2+ Y^2-3)\in \mathbf{R}[X,Y]$ ("Motzkin polynomial"). (a) One has $M(x, y)\geqslant 0$ for all $x$ and $y$ in $\mathbf{R}$.

(b) There exists no finite family $(p_i)_{i\in I}$ in $\mathbf{R}[X,Y]$ such that $M =\sum p^2_i$. (If such a family existed, now prove that each $p_i$ must be of the form $a+b$XY with $a\in \mathbf{R}$ and $b\in \mathbf{R}[X,Y]$ of degree at most 1; then calculate the coefficient of (XY)$^2.$)

(c) Let $\tau :\mathbf{N}^2\rightarrow \mathbf{N}-\{0\}$ be the mapping defined by

and

$\tau (i, j) =j+ 1 + (i+j)(i+j+ 1)/2$ if $i+j\geqslant 4$.

The mapping $\tau$ is a bijection.

(d) Let $(\nu_n)_{n\geqslant 1}$ be the sequence defined by

$\nu_i= 1$ if $i\in \{1,2,3\},\nu_4= 4,\nu_n= (n$!)$^{(n+1)!}$ if $n\geqslant 5$.

For $n$ and $m$ in $\mathbf{N}$, put $\mu_{n,m}$ = 0 if $n$ or $m$ is odd and $\mu_{n,m}=\nu_{\tau(n/2,m/2)}$ otherwise. The linear mapping $\alpha$ of $\mathbf{C}[X,Y]$ into $\mathbf{C}$ such that $\alpha (X^nY^m) =\mu_{n,m}$ satisfies $\alpha (p^2)\geqslant 0$ for every $p\in \mathbf{R}[X,Y]$. (It is enough to verify that, for every integer $k\in \mathbf{N}$, the determinant of the matrix $(a_{m,m})_{1\leqslant n,m\leqslant k}$ is positive, where $a_{n,m}$ is defined by $a_{\tau(i,j),\tau(i',j')}=\alpha (X^{i+i'}Y^{j+j'})$ for all $(i, j, i', j')$; then proceed by induction on $k.$)

(e) There exists no positive measure $\nu$ on $\mathbf{R}^2$ such that $(x, y)\mapsto x^ny^m$ is $\mu$-integrable for all $(n, m)\in \mathbf{N}^2$, and such that

for all $(n, m)\in \mathbf{N}^2$. (f) There exist a Hilbert space E and a partial operator $u$ with dense domain on E satisfying the following conditions:

(i) One has dom($u$)$\subset$ dom($u^*$);

(ii) One has $\|u(x)\|=\|u^*(x)\|$ for every $x\in$ dom($u$);

(iii) There exist no Hilbert space F containing E as a subspace, such that the canonical injection of E into F is isometric, and no normal partial operator $v$ on F such that $u\subset v$.

Compare this result with exercise 24 of IV, p. 352.