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 in Chalmoukis (Proc Am Math Soc 148(8):3325–3337, 2020) 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 \(H^q, \, 0<p,q<\infty .\)
