  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.
