Fractional operators as traces of operator-valued curves

We relate non integer powers ${\mathcal{L}}^s$, $s>0$ of a given (unbounded) positive self-adjoint operator $\mathcal{L}$ in a real separable Hilbert space $\mathcal{H}$ with a certain differential operator of order $2\lceil s\rceil$, acting on even curves $\mathbb{R}\to \mathcal{H}$. This extends the results by Caffarelli–Silvestre and Stinga–Torrea regarding the characterization of fractional powers of differential operators via an extension problem.