  The Fock space can be characterized (up to a positive multiplicative factor)
as the only Hilbert space of entire functions in which the adjoint of
derivation is multiplication by the complex variable. Similarly (and still up
to a positive multiplicative factor) the Hardy space is the only space of
functions analytic in the open unit disk for which the adjoint of the backward
shift operator is the multiplication operator. In the present paper we
characterize the Hardy space in term of the adjoint of the differentiation
operator. We use reproducing kernel methods, which seem to also give a new
characterization of the Fock space.
