  We consider operators of the type $D^\alpha:H^2(\mathcal{H})\to
H^2(\mathcal{H})$, where $D^\alpha$ denotes a fractional differentiation
operator, and $\Gamma_\phi$ is a Hankel operator. For $\alpha>0$, we
characterize boundedness in terms of a natural anti-analytic Carleson embedding
condition. We obtain three notable corollaries. The first is that our main
result does not extend to $\alpha=0$, i.e. Nehari-Page BMOA is not
characterized by the natural anti-analytic Carleson embedding condition. The
second is that when we add an adjoint embedding condition, we obtain a
sufficient but not necessary condition for boundedness of $\Gamma_\phi$. The
third is that there exists a bounded analytic function for which the associated
anti-analytic Carleson embedding is unbounded. As a consequence, boundedness of
an analytic Carleson embedding does not imply that the anti-analytic ditto is
bounded. This answers a question by Nazarov, Pisier, Treil, and Volberg.
