I find myself studying the spectral theorem but fail to understand the following claim I just saw in an article.
Let $A:=\sqrt{-\Delta+1}$ be an operator. The operator $A$ on $L^2$ with domain $H^1$ (Sobolev space) is self-adjoint. Moreover its extension to $H^{1/2}$, which is denoted by $A:H^{1/2}\to H^{-1/2}$, generates a $\mathcal{C}^{0}$-group of isometries, $\left\{\mathrm{e}^{-itA}\right\}_{t\in\mathbb{R}}$, acting on $H^{1/2}$.
From what I see, since $L^2=H^{0}\subset H^{-1/2}$, then it makes sense to talk about an operator extension to a larger space.
Question 1: Why can the operator be extended to a larger space?
Question 2: Why does the extension turn out to be defined on the space $H^{-1/2}$?
What I know so far is that the Laplace operator is a self-adjoint operator in its maximal domain (Simon, Reed, Math Physics vol 2), but I can't understand why the operator $A$ can be extended.
Thank you very much.