Integration operators on Hardy and Bergman spaces

In the present work, we are interested in compact integration operators $I_{g}f(z)={\int }_0^{z}f(\zeta )g^{\mathrm{\prime }}(\zeta )d\zeta$ acting on the Hardy space $H^2$ and on the weighted Bergman spaces ${\mathcal{A}}_{\alpha }^2$. We give upper and lower estimates for the singular values of $I_g$.