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.

In a separable Hilbert space H, two frames {f i } i∈I and {g i } i∈I are said to be woven if there are constants 0 < A ≤ B so that for every σ ⊂ I,This article provides methods of constructing woven frames. In particular, bounded linear operators are used to create woven frames from a given frame.Several examples are discussed to validate the results. Moreover, the notion of woven frame sequences is introduced and characterized.

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In frame theory literature, there are several generalizations of frame, K-fusion frame presents a flavour of one such generalization, basically it is an intertwined replica of K-frame and fusion frame. Kfusion frames come naturally (having significant applications) when one needs to reconstruct functions (signals) from a large data in the range of a bounded linear operator. Getting inspiration from the concept of weaving frames in Hilbert space, we study the weaving form of Kfusion frames which have significant applications in wireless sensor networks. This article produces various characterizations of weaving Kfusion frames in different spaces. Furthermore, Paley-Wiener type perturbation and conditions on erasure of frame components have been assembled to scrutinize woven-ness of the same.