Exercise 21, § 3

We retain the notation of the preceding exercise.

a) Let $f\in \mathscr{L}^2(G)$ be of positive type as a locally integrable function. The partial operator on $L^2(G)$ with domain $\mathscr{K}(G)$ defined by $\varphi \mapsto$ $\boldsymbol{\gamma }(\varphi )f$ is positive. We denote by $\varrho (f)$ the Friedrichs extension of $f$ (example 8 of IV, p. 295); it is a positive self-adjoint partial operator on $L^2(G)$.

b) Let $f\in \mathscr{L}^2(G)$. One says that $f$ is sober if there exists a real number $c\geqslant 0$ such that $\|\boldsymbol{\gamma }(\varphi )f\|\leqslant c\|\varphi \|$ for every $\varphi \in \mathscr{K}(G)\subset \mathscr{L}^2(G)$. One then denotes by $\varrho (f)$ the endomorphism of $L^2(G)$ extending $\varphi \mapsto \boldsymbol{\gamma }(\varphi )f$. This definition coincides with the preceding one if $f$ is of positive type, and this condition then amounts to saying that $\varrho (f)$ is positive.

c) Let $f_1$ and $f_2$ be sober functions in $\mathscr{L}^2(G)$. If $\varrho (f_1) =\varrho (f_2)$, then $f_1$ and $f_2$ are equal in $L^2(G)$.

d) Let $f\in \mathscr{L}^2(G)$ be of positive type. For every $g\in G$, one has $\varrho (f)\circ \boldsymbol{\gamma }(g) =$ $\boldsymbol{\gamma }(g)\circ \varrho (f)$. Moreover for every $h\in$ dom($\varrho (f)$), one has $\varrho (f)h=h*f$.

e) Let $f_1$ and $f_2$ be sober elements of $\mathscr{L}^2(G)$ such that $\varrho (f_1)$ and $\varrho (f_2)$ commute. The function $f_1*f_2$ is a continuous function of positive type on G; it is sober, and one has

f) With the notation and hypotheses of the preceding question, if $f_2-f_1$ is of positive type, then

g) Let $(f_n)_{n\in\mathbf{N}}$ be a sequence of sober positive definite elements of $\mathscr{L}^2$(G), pairwise permutable, and such that $f_{n+1}-f_n$ is of positive type for every $n\in \mathbf{N}$. If the sequence $(\|f\|_n)$ is bounded, then the sequence $(f_n)$ converges in $\mathscr{L}^2(G)$.