How to prove the following using Hilbert Scales?

Question

If $L$ is densely defined strictly positive, unbounded, selfadjoint operator in Hilbert space $X$. We consider $M$ be the set of all elements $x$ for which all the powers of $Lx$ are defined, i.e. $$M = \bigcap\limits_{k = 0}^{\infty} D(L^k)$$Then how to prove that $L^s$ is defined on $M$ for all $s \in \mathbb{R}$ and $$M = \bigcap\limits_{s \in \mathbb{R}} D(L^s)$$ My iind question is can we talk of completeness of $M$ with some norm?