On the domains of fractional powers of a selfadjoint operator

Question

Let $T$ be an unbounded selfadjoint positive operator on an Hilbert space ( say $L^2(\mathbb{R}^3)$ ). By Borel functional calculus, i can define the powers $T^r$ for r>0. Suppose that i know the action of $T^r$ on a generic element of its domain, and that this action can be extended to a generic $L^2$ function (taking value in the space of tempered distribution). I still call this map $T^r$ (improperly). Is it true that $$f\in L^2, T^rf\in L^2\Rightarrow f\in\mathcal{D}(T^r)$$