Is there any simple way to characterize explicitly the domain of fractional powers for a given operator? For example, the domain of Dirichlet Laplacian on a bounded nice domain $\Omega \subset \mathbb{R}^n$: $$(-\Delta)^\alpha, \quad \alpha \in (0,1).$$ I know that there is a characterization using interpolation theory, so one can find $H^{2\alpha}(\Omega)$ or $H^{2\alpha}_0(\Omega)$. But I'm asking if there is a systematic and simple way to do that, so we can use it for more complicated operators, for example matrix valued operators.

Any reference which gives the proof of the above characterization or a related topic will be very helpful.

Domains, Uniqueness and the Cauchy Problemin [Martínez, Sanz,The Theory of Fractional Powers of Operators, Elsevier, 2001] might be a good starting point. $\endgroup$