K-frames And K-duals In Hilbert Spaces

K-frames in Hilbert spaces are the generalizations of frames and there are many differences between K-frames and ordinary frames. In this thesis, we discuss the relation between a K-frame and the operator K, and characterize the optimal bounds of a K-frame by using the synthesis operator and the operator K. We introduce the definition of K-dual of a K-frame and obtain some properties of K-duals. Also, the uniqueness of the K-dual for a Parseval K-frame is investigated. Then we discuss some properties about two K-frames in Hilbert space from which we can get new K-frames or Parseval K-frames under the conditions of orthogonality, disjointness or other special conditions; moreover, we get all the K-duals for Parseval K-frames with the special K-duals for Parseval K-frames, and get K-duals which are Parseval *K-frames by the orthogonality. In the end we investigate the applications of K-frames in finite dimensional Hilbert spaces utilizing the K-duals. Especially, we discuss the relations of K-frames, K-duals, and the traces of the relevant operators. In addition, we seek for the K-duals of Parseval K-frames that are the optimal for erasures. We will give some necessary and sufficient conditions under which the canonical K-dual is the unique optimal K-dual for erasures. We still discuss some special conditions under which the canonical K-dual is either not the optimal or it is the optimal K-dual but not the unique one.