On some operators and dilations of frame generator and dual pair of frame generators of two structured unitary systems

In this paper, we study the dilations of frame generators and pairs of dual frame generators of two special structured unitary systems for Hilbert spaces, i.e. the product unitary system and group-like unitary system. We first introduce certain operators which are associated with the frame generators of the unitary system and some properties of the operators are established. Then, as an application, we show that the canonical dual of a frame of group-like unitary systems remain the group-like structure. And we use these properties to study the dilations of the frame generators and dual pairs of frame generators of these two types of unitary systems.