Let $H$ Hilbert space and $B$ a strictly positive self-adjoint, bounded operator on $H$. Now fix $\alpha>0$ then we can define the fractional powers $B^{\alpha}$ which are strictly positive self-adjoint, bounded operators on $H$.
Now define by $H_{-\alpha}$ the the completion of $H$ in the norm $$ \|x\|_{-\alpha}=\left\langle B^{\alpha/2} x, B^{\alpha/2}x\right\rangle^{1 / 2} $$ (observe $\|x\|_{-\alpha}$ is weaker than $\|x\|_H$) and then it is a Hilbert space. Now in [Crandall, Michael G., and Pierre-Louis Lions. "Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. V. Unbounded linear terms and B-continuous solutions." Journal of functional analysis 97.2 (1991): 417-465.] I read "in the natural way, $B^{\alpha / 2}$ can be extended to an isometry from $H_{-\alpha}$ onto $H$".
Why can I extend $B^{\alpha / 2}$ to an isometry from $H_{-\alpha}$ onto $H$? In particular take $x \in H_{-\alpha}$ then there exist $x_n \in X$ such that $|B^{\alpha/2} (x_n-x)|\to0$ so that I can define $B^{\alpha/2}x=\lim_n B^{\alpha/2}x_n$. Now since $H_{-\alpha}$ is the completion of $H$ then
$B^{\alpha/2}x_n$ is cauchy in $H$ and then $B^{\alpha/2}x \in H$ since $H$ is HIlbert and hence complete. Is this right?
