[Objective]The K-frame is a more general kind of frame related to a bounded linear operator K in Hilbert spaces,it was first proposed by Gavruta L.in 2013 for the purpose of studying the atomic decompositions in Hilbert spaces.It is the combination of the K-frame with a bounded linear operator K that gives the K-frame many unique properties compared to the classical frames.It is well known that the pair of sequences of an alternate dual of classical frames are woven,in this paper we explore whether the two involved two sequences of a K-dual with K-frames are woven.At the same time,we discuss several characterizations of the eigenvalues of the frame operator of the K-frames.[Methods]We will study the weaving of the pair of sequences of a K-dual and the characterizations of the eigenvalues of the frame operator of the K-frames with the help of tools such as operator norm,sequence summation,orthogonality of the the operator theory of Banach spaces and eigenvalues in algebra.[Results]The two involved two sequences of a K-dual with K-frames are generally not woven in the whole space.Then we construct a new pair of sequences based on the two sequences involved in the K-dual,making them K-woven in H.Two characterizations of eigenvalues and corresponding eigenvectors of K-frame and classical frame are obtained.[Conclusions]Compared with classical frames,we get more general conclusions about the weaving of two sequences involved in the K-duality of K-frames and the characterizations of the eigenvalues of the frame operators of K-frames.When K is the identity operator of Hilbert space,the obtained K-frame conclusions in this paper are the corresponding results of classical frames.
万方数据
维普
