Atomic subspaces for operators

Abstract

This paper introduces the concept of atomic subspaces with respect to a bounded linear operator. Atomic subspaces generalize fusion frames and this generalization leads to the notion of K-fusion frames. Characterizations of K-fusion frames are discussed. Various properties of K-fusion frames, for example, direct sum, intersection, are studied.

Generalized atomic subspaces for operators in Hilbert spaces

“…Definition 2.6. [1] Let { W j } j ∈ J be a family of closed subspaces of H and { v j } j ∈ J be a family of positive weights and K ∈ B ( H ). Then { ( W j , v j ) : j ∈ J } is said to be an atomic subspace of H with respect to K if following conditions hold:…”

“…P. Casazza [2] was first to introduce the notion of fusion frames or frames of subspaces and gave various ways to obtain a resolution of the identity operator from a fuison frame. The concept of an atomic subspace with respect to a bounded linear operator were introduced by A. Bhandari and S. Mukherjee [1]. Construction of K-g-fusion frames and their dual were presented by Sadri and Rahimi [14] to generalize the theory of K-frame, fusion frame and g-frame.…”

Perturbations on K-fusion frames

“…K-fusion frame. In [3], authors, introduced a generalization of fusion frame, K-fusion frame, and scrutinized the equivalence between atomic subspaces and K-fusion frames. Kfusion frame is used to reconstruct signals from range of a bounded linear operator K.…”

“…Frame theory literature became richer through several generalizations-fusion frame (frames of subspaces) [4,6] , G-frame (generalized frames) [22], K-frame (atomic systems) [14], K-fusion frame (atomic subspaces) [3], etc. and these generalizations have been proved to be useful in various applications.…”

Characterizations of woven frames

“…Frame theory literature became richer through several generalizations, namely, G-frame (generalized frames) [3], K-frame (frames for operators (atomic systems)) [4], fusion frame (frames of subspaces) ( [5,6]), K-fusion frame (atomic subspaces) [7], etc. and some spin-off applications by means of Gabor analysis in ( [8,9]), dynamical system in mathematical physics in [10], nature of shift invariant spaces on the Heisenberg group in [11], characterizations of discrete wavelet frames in C N in [12], extensions of dual wavelet frames in [13], constructions of disc wavelets in [14], orthogonality of frames on locally compact abelian groups in [15] and many more.…”