Canonical dual K-Bessel sequences and dual K-Bessel generators for unitary systems of Hilbert spaces

In this paper, firstly we explore some properties of K-frames in terms of decompositions of operators. Then we pay more attention to characterize the dual K-Bessel sequences of a given K-frames. Since the frame operator for a K-frame may not be invertible, there is no classical canonical dual for a K-frame. We introduce the concept of canonical dual K-Bessel sequence of a K-frame which generalizes the classical dual of a frame. We prove its existence and uniqueness and some other properties. We also give a simple way to construct new K-frames from given ones. For the K-frame generators of a unitary system, we give a condition such that a K-frame generator for a unitary group has a dual K-Bessel generator so that both of them share the same group structure. We characterize the K-frame generators in terms of operators and we also consider the construction of new K-frame generators from given ones.