  We study the boundedness and compactness properties of the generalized
integration operator $T_{g,a}$ when it acts between distinct Hardy spaces in
the unit disc of the complex plane. This operator has been introduced by the
first author in connection to a theorem of Cohn about factorization of higher
order derivatives of functions in Hardy spaces. We answer in the affirmative a
conjecture stated in the same work, therefore giving a complete
characterization of the class of symbols $g$ for which the operator is bounded
from the Hardy space $H^p$ to $H^q, \, 0<p,q<\infty.$
