Closed range weighted composition operators and dynamical sampling

Abstract

We solve the closed range problem for weighted composition operators on Fock spaces. The result equivalently characterizes when the operators are bounded from below. We give several applications of the main result related to the operators invertibility, Fredholm, and dynamical sampling structures from frame perspectives. We prove there exists no vector in the Fock space for which its orbit under the weighted composition operator represents a frame family. Furthermore, it is shown that a weighted composition operator preserves frames if and only if it preserves the stronger Riesz basis property. Similar results are provided for the adjoint operator.