Continuous K-g-Frames in Hilbert Spaces

Abstract

In this paper, we intend to introduce the concept of c-K-gframes, which are the generalization of K-g-frames. In addition, we prove some new results on c-K-g-frames in Hilbert spaces. Moreover, we define the related operators of c-K-g frames. Then, we give necessary and sufficient conditions on c-K-g-frames to characterize them. Finally, we verify perturbation of c-K-g-frames.2010 Mathematics Subject Classification. Primary 42C15, 42C40.

Frames and weak frames for unbounded operators

“…In this section we introduce our second approach to the generalization of the notion of (discrete) atomic system for K ∈ B(H) and of K-frame in [20], to unbounded operators in a Hilbert space in the continuous framework. Since a closed densely defined operator in a Hilbert space A : D(A) → H can be seen as a bounded operator A : H A → H between two different Hilbert spaces (with H A the Hilbert space D(A)[ · A ] where · A is the graph norm), before introducing new notions, we put the main definitions and results in [2,20] for K ∈ B(H) in terms of bounded operators from a Hilbert space into another. Later, in Section 4.1, we return to the operator A : H A → H.…”

“…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].…”

“…In order to decompose the range of a densely defined unbounded operator A as a combination of vectors in H, we need somewhat which takes on its unboundedness. In literature there are some generalizations to the continuous case of the notion of K-frame (as, e.g., c-K-g-frames in [2]), however, as far as the author knows, the case of an unbounded operator K in H has been little considered.…”

Continuous frames for unbounded operators

“…K-frames allow to write each element of RðKÞ, 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 RðKÞ with K 2 BðHÞ. K-frames have been generalized in [4] and [23] where the notion of K-gframes was investigated and have been further generalized in 2018 to the continuous case in [2].…”

“…In literature there are some generalizations to the continuous case of the notion of K-frame, as e.g. c-K-gframes in [2]); however, as far as the author knows, the case of an unbounded operator K in H has been little considered.…”