Topological and iterated structures of the superposition operator on Fock spaces

Abstract We characterize various topological structures of the weighted superposition operator on Fock spaces. The global homeomorphism property of the operator is described by its ray-invertibility condition. We further consider its iterated structures and prove that the operator is power bounded if and only if it is mean ergodic. However, Fock spaces do not support cyclic weighted superposition operators, and as a result, no orbit of the operator forms a frame. We also identify a condition under which the operator preserves frames and Riesz bases. Finally, we examine the semigroups generated by the operator. In the end, we investigate the semigroups of the operator. This work provides a systematic investigation of a wide range of properties of the operator, and can serve as a valuable exposition, illustrating the interplay between fundamental operator-theoretic concepts in both linear and nonlinear analysis.