Some Closed Range Integral Operators on Spaces of Analytic Functions

Abstract

Our main result is a characterization of g for which the operator \({S_g(f)(z) = \int_0^z f'(w)g(w)\, dw}\) is bounded below on the Bloch space. We point out analogous results for the Hardy space H2 and the Bergman spaces Ap for 1 ≤ p < ∞. We also show the companion operator \({T_g(f)(z) = \int_0^z f(w)g'(w) \, dw}\) is never bounded below on H2, Bloch, nor BMOA, but may be bounded below on Ap.