I recently stumbled on the following equality:
$$ \| (-\frac{d^2}{dx^2} + 1)^{1/2}g\|_{L^2} = \| g\|_{H^1}$$ for $g \in H^2(\mathbb{R})$. I tried to deduce the equality but failed (since I don't really have an idea how to attack this problem), and I also wasn't able to find a proof. If anyone would share a hint with me on how to approach, I would appreciate it!
You have $$\| (-\frac{d^2}{dx^2} +1)^{1/2} g\|_{L^2}^2 = \int_{\mathbb{R}} g \cdot (-\frac{d^2}{dx^2} +1) g \, dx = \int_{\mathbb{R}} g^2 + (\frac{d}{dx} g)^2 \, dx = \|g\|_{H^1}^2,$$ where we used integration by parts in the second to last step.