K-G-Frames and Stability of K-G-Frames in Hilbert Spaces

Abstract

Abstract. A K-g-frame is a generalization of a g-frame. It can be used to reconstruct elements from the range of a bounded linear operator K in Hilbert spaces. K-g-frames have a certain advantage compared with g-frames in practical applications. In this paper, the interchangeability of two g-Bessel sequences with respect to a K-g-frame, which is different from a g-frame, is discussed. Several construction methods of K-g-frames are also proposed. Finally, by means of the methods and techniques in frame theory, several results of the stability of K-g-frames are obtained.

Continuous K-g-Frames in Hilbert Spaces

“…Kframes allow us in a stable way, to reconstruct elements from the range of a linear and bounded operator in a Hilbert space. K-g-frames have been introduced in [3,11] and some properties and characterizations of K-g-frames has been obtained, for more information on K-gframes, the reader can check [11,20]. In this paper, we generalize some results in [11] and [20] to continuous version of frames.…”

“…Perturbation of discrete frames and frames associated with measurable spaces (c-frame) have been discussed in [5] and [9], respectively, and perturbations of g-frames, c-g-frames and their dual have been discussed in [14], [21] and [1]. Stability and perturbation of K-g-frames have been investigated in [3], [22,23] and [11]. In this section, we introduce perturbation of c-K-g-frames.…”

K-g-fusion frames in Hilbert spaces

“…Stability of a frame is an important research topic for frame theory. Many researchers discussed the stability of frames, fusion frames, K -frames, and K -g-frames [3,7,8,13,14,16]. In the following theorems, we give some results about perturbations of K -g-fusion frame.…”

Frames and weak frames for unbounded operators

“…K-frames allow to write each element of R(K), the range of K, which is not a closed subspace in general, as a combination of the elements of the K-frame, which do not necessarily belong to R(K) with K ∈ B(H). K-frames have been generalized in [4] and [21] where the notion of K-g-frames was investigated and have been further generalized in 2018 to the continuous case in [2].…”