Some Closed Range Integral Operators On Spaces of Analytic Functions

Our main result is a characterization of $g$ for which the operator $S_g(f)(z)={\int }_0^z{f}^{\mathrm{\prime }}(w)g(w)dw$ is bounded below on the Bloch space. We point out analogous results for the Hardy space $H^2$ and the Bergman spaces $A^p$ for $1\le p<\mathrm{\infty }$. We also show the companion operator $T_g(f)(z)={\int }_0^{z}f(w)g^{\mathrm{\prime }}(w)dw$ is never bounded below on $H^2$, Bloch, nor BMOA, but may be bounded below on $A^p$.