TITLE: Fractional powers of positive self-adjoint operators
QUESTION [6 upvotes]: Consider two positive unbounded operators $A$ and $B$ densely defined on a Hilbert space $H$ self-adjoint on a domain $\mathcal{D}(A) = \mathcal{D}(B) = H_1$. By the spectral theorem, we can define the fractional powers of $A$ and $B$ as self-adjoint linear operators on $H$. My question is, is $\mathcal{D}(A^{\alpha}) = \mathcal{D}(B^\alpha)$, where $\alpha \in (0, 1)$ necessarily? Is this part of the spectral theorem? If yes, where can I find such a statement?

REPLY [2 votes]: If $A$ and $B$ are positive definite densely-defined selfadjoint linear operators on a complex Hilbert space $H$, then $AB^{-1}$ and $BA^{-1}$ are bounded if $\mathcal{D}(A)=\mathcal{D}(B)$. Then $A^{s}B^{-s}$ and $B^{s}A^{-s}$ are bounded for $0 < s \le 1$ with
$$
                \|A^{s}B^{-s}\| \le \|AB^{-1}\|^{s} \\
                \|B^{s}A^{-s}\| \le \|BA^{-1}\|^{s}.
$$
This implies that $\mathcal{D}(A^{s})=\mathcal{D}(B^{s})$ for $0 < s \le 1$.
Reference (Inequality a3): https://www.encyclopediaofmath.org/index.php/Heinz-Kato_inequality