  For g in BMOA, we consider the generalized Volterra operator T_g acting on
Hardy spaces H^p. This article aims to study the largest space of analytic
functions, which is mapped by T_g into the Hardy space H^p. We call this space
the optimal domain of T_g and we describe its structural properties. Motivation
for this comes from the work of G. Curbera and W. Ricker who studied the
optimal domain of the classical Ces\'aro operator.
