Frames for operators

Continuous Atomic Systems for Subspaces

“…For basic results on frames, we refer the reader to the references [10,12,14,21,22,29,30]. Here, it should be noted that H which satisfied in the equality (2) is important in the theory of frames and in many situations, the study of such family of operators is of considerable importance; see for example, [3,[5][6][7]9,11,19,20,23,25,27,28,31,32,[34][35][36][37][38][39][40]42,43]. Therefore, it is expected that the study of the family of bounded operators which satisfied in the equality (3) plays a crucial role in the theory of frames in Hilbert space.…”

“…The motivation comes from some recent developments of the theory of frames in Hilbert spaces and the problems arising in sampling theory which has been studied mostly by Feichtinger and his coauthors in [16][17][18][19]23]; see also [24,27] for recent work. More precisely, we introduce and study the concept of continuous atomic systems for subspaces, and we give some examples to show differences between this new notion and the discrete version.…”

Frame sequences and dual frames for operators

“…From Ref. 9 we know that K-frames possess higher generality than ordinary frames in the sense that the lower frame bound condition holds only for the elements in the range of K and that they allow the reconstruction of the elements from the range of K in a stable way and, in general, the range is not even a closed space. Hence Kframes provide more flexibility and thus make the study of them interesting.…”

“…Gȃvruţa 9 recently presented a generalization of ordinary frames with a linear bounded operator K, named K-frames, when working on atomic systems for operators. From Ref.…”

Frames for B(ℋ)

“…The notion of Operator frame for the space B(H ) of all bounded linear operators on Hilbert space H was introduced by Chun-Yan Li and Huai-Xin Cao [1] and the notion of Kframe for an operator K ∈ B(H ) was introduced by L.Guvruta [10]. In this paper, we consider the fusion of the two concepts and introduce K -operator frame as a generalisation of both Kframe and operator frame for B(H ) and obtain some results which are more general than the results proved in [1] and [10]. K -dual of a K -operator frame for B(H ) is also introduced.…”