Trace ideal criteria for generalized integration operators

For $g \in \operatorname{Hol}(\mathbb D)$, we study the class of generalized integration operators $T_{g,a}$, acting on Hardy and Bergman spaces of the unit disc in the complex plane. This class of integral operators were introduced to study factorization theorems in the Hardy spaces of the unit disc. We completely characterize the space of symbols g, for which $T_{g,a}$ belongs to the Schatten-von Neumann ideals of the Hardy and Bergman spaces.