Consider Soblev spaces as special cases of Besov spaces characterized by wavelets, $H^s = B^s_{2,2}$.
I want to prove the following statement. For a positive definite self-adjoint operator $A$, if the following satisfies for any $f\in H^s$ with $s>0$,
$$\langle Af, f\rangle \simeq \|f\|_{H^{-t/2}},$$
where $\simeq$ means equality holds up to an independent constant of $f$ and $t>0$, then $A$ has the lifting property such that $A:H^s \to H^{s+t}$.
It seems to be a rather standard property used in literature, e.g. this and this.