Let $D$ be a self-adjoint (in the $H^0$-inner product) first-order differential operator on a manifold $M$, where $H^i$ stands for the $i$-th Sobolev space on $M$. Then $D$ extends to a bounded operator $H^{i+1}\rightarrow H^{i}$ for each $i\geq 0$, self-adjoint in the $i$-norm. To distinguish between norms, let us write $(D^2+1)_i$ for $D^2+1:H^{i+2}\rightarrow H^i$. 

Then for each $i\geq 0$, $(D^2+1)_i$ has a bounded positive inverse

$$(D^2+1)_i^{-1}:H^i\rightarrow H^{i+2}.$$

Thus for every $i$ we get a unique positive square root $(D^2+1)_i^{-1/2}$, which is a bounded operator $H^i\rightarrow H^i$. The range of this square root can be shown to be $H^{i+1}$. I have a number of (related) questions about these operators.

**Question 0:** Is $(D^2+1)^{-1/2}_i$ a bounded operator $H^i\rightarrow H^{i+1}$?

**Question 1:** Do $(D^2+1)_i^{-1/2}$ and $(D^2+1)_j^{-1/2}$ for $i\neq j$ coincide on their common domain $H^{\text{max}\{i,j\}}$?

**Question 2:** Is the domain of the unique (unbounded) positive square root of $(D^2+1)_i$ equal to $H^{i+1}$? Let us call this square root $(D^2+1)_i^{1/2}.$

**Question 3:** Is $(D^2+1)_i^{-1/2}$ the inverse of $(D^2+1)_i^{1/2}$?

Thanks for your help.