Lemma 8

Let E be a separated topological vector space. Let $(E_1,\|\cdot \|_1)$ and $(E_2,\|\cdot \|_2)$ be dense subspaces of E endowed with Hilbert space structures such that the canonical injections of $E_1$ and $E_2$ into E are continuous. Let F be a subspace of $E_1\cap E_2$, dense in $E_1$ and $E_2$ for these Hilbert space structures. If $\|x\|_1=\|x\|_2$ for every $x\in F$, then $E_1= E_2$ and $\|\cdot \|_1=\|\cdot \|_2$.

Let $x\in E_1$. There exists a sequence $(x_n)_{n\in\mathbf{N}}$ in F such that $x_n$ converges to $x$ in the Hilbert space $E_1$. Since $\|x_n-x_m\|_2=\|x_n-x_m\|_1$ for all integers $n$ and $m$, the sequence $(x_n)$ is a Cauchy sequence in $E_2$. Let $y$ be its limit. We have $\|y\|_2=$ lim$\|x_n\|_2=\|x\|_1$. Since the canonical injections of $E_1$ and $E_2$ into E are continuous, we have $x_n\rightarrow x$ in E and analogously $x_n\rightarrow y$ in E. Thus it follows that $x=y$ and $\|x\|_1=\|x\|_2$; in particular, $E_1\subset E_2$. We conclude by symmetry.

Let E be a complex Hilbert space. Let $u$ be a self-adjoint positive partial operator on E. For every $\alpha \in \mathbf{R}_+$, we set $u^{\alpha}=f(u)$, where $f$ is the continuous mapping $x\mapsto x^{\alpha}$ of $\mathbf{R}_+$ into $\mathbf{R}$. This is a self-adjoint positive partial operator (Cor. 1, a) of IV, p. 273). When $u\in \mathscr{L}$ (E), this notation is compatible with the notation relative to the involutive algebra $\mathscr{L}(E) ($cf. Prop. 16 of I, p. 118). If $\alpha$ is a positive integer, the partial operator $u^{\alpha}$ coincides with the partial operator defined by the composition $u\circ \cdots \circ u$ (Cor. 1, b) of IV, p. 273).

Let $\beta \in \mathbf{R}_+$. We have $u^{\alpha \beta}= (u^{\alpha})^{\beta}$ (Cor. 4 of IV, p. 274). In particular, if $\alpha >0$, then the partial operator $u^{1/\alpha}$ is the unique self-adjoint positive partial operator $v$ on E such that $v^{\alpha}=u$.

Suppose that $0\leqslant \alpha \leqslant \beta$. Then dom($u^{\beta}$)$\subset$ dom($u^{\alpha}$) $:$ in fact, for every real number $x\geqslant 0$, one has $x^{\alpha}\leqslant 1 +x^{\beta}$, and the result then follows from the definition of the domain of $u^{\alpha}($cf. Prop. 3 of IV, p. 271).

Let $u$ be a closed operator with dense domain from E into a complex Hilbert space F. The partial operator $u^*u$ is self-adjoint and positive (Prop. 12 of IV, p. 241). We write $|u|= (u^*u)^{1/2}$.