Some integral operators acting on H

Let f and g be analytic on the unit disk D . The integral operator T g is defined by T g f ( z ) = ∫ 0 z f ( t ) g ′ ( t ) d t , z ∈ D . The problem considered is characterizing those symbols g for which T g acting on H ∞ , the space of bounded analytic functions on D , is bounded or compact. When the symbol is univalent, these become questions in univalent function theory. The corresponding problems for the companion operator, S g f ( z ) = ∫ 0 z f ′ ( t ) g ( t ) d t , acting on H ∞ are also studied.