K-frames

K-g-frames, as an extension of g-frames and K-frames are one of the active fields in frame theory. In this paper, we  consider continuous K-g-frames which are a generalization of discrete K-g-frames. We give the necessary and sufficient conditions to characterize their duals. For example, the canonical dual  of a given K-g-frame is described by both its frame operator and its alternate duals.

In this paper, we reintroduce the concept of controlled K-frames and then, we show that this definition is equivalent with the concept that has been recently introduced in cite{Nouri}. Meanwhile, we correct one of the results which were obtained in the mentioned paper. In the sequel, we construct some new controlled K-frames by some operator theory tools. Finally, we provide some conditions under which the sum of two controlled K-frames remains a controlled K-frame.