Closed Range Integral Operators on Fock Spaces

We study the closed range problem for generalized Volterra-type integral operators on Fock spaces. We first answer the problem using the notions of sampling sets, reverse Fock–Carleson measures, Berezin type integral transforms, and essential boundedness from below of some functions of the symbols of the operators. The answer is further analyzed to show that the operators have closed ranges only when the derivative of the composition symbol belongs to the unit circle. It turns out that there exists no nontrivial closed range integral operator acting between two different Fock spaces. The main results equivalently describe when the operators are bounded below. Explicit expressions for the range of the operators are also provided, namely that the closed ranges contain only elements of the space which vanish at the origin. We further describe conditions under which the operators admit order bounded structures.