Exact K-frames In Hilbert Spaces

The concept of frame in Hilbert spaces was firstly introduced by Duffin and Schaeffer in 1952 to study nonharmonic Fourier series. In fact, a frame is a generalization of the orthonormal basis and has non-unique representation of space elements in terms of frame. This nature makes it become a very useful tool in practice. With the development of the theory of the frame, some scholars put forward a series of new generalized frames such as bounded quasi-projector, pseudo-frames, oblique frame, outer frame, g-frames and so on, which have a lot of help in practical application. In 2010 Gavruta proposed the K-frame, which had a close relationship with atomic system, was a more general frame than the frame in Hilbert spaces. The only difference between a K-frame and a frame lies in the lower bound of the K-frame. Although K-frames and frames have a lot of similar properties, not all of the properties are similar. For example, l2 linearly independent frames are equivalent to exact frames, but I2 linearly independent K-frames are not equivalent to exact K-frames. Which properties of the frame may be extended to the K-frame for a complex Hilbert space? In this paper, on the basis of the theory of frames, we study equivalent relationship of exact K-frame, its depiction of operator and stability. We generally arrange the text structure as follows:In chapter 1. we make a brief introduction about the frame’s emergence and its development.In chapter 2, we review simply some basic concepts and important properties of the frame and introduce mainly several important properties of K-frames and g-frames. Then, we give the main content about this article.In chapter 3, although an l2 linearly independent K-frame and an exact K-frame are inequivalence, we can find an equivalent relationship between an l2 linearly independent K-frame and an exact K-frame, and give the equivalent condition. Then, we discuss depictions of operators under K operator of K-frame and T bounded linear operator in Hilbert spaces.In chapter 4, firstly, we study that a K-frame is a frame under certain operator, and we give some conditions that a K-frame is a frame. We can indicate that the synthesis operator of K-frame is not necessarily closed. Secondly, we mainly discuss the stability of exact g-frames under perturbation according to the equivalent relationship between an exact K-frame and an l2 linearly independent K-frame. Meanwhile, we also use the theory of operator in functional analysis to study stability of exact K-frames.