GANITA, Vol. 67(1), 2017, 41-52 41 Weaving K -Fusion Frames in Hilbert Spaces Saakshi Garg & Lalit K. Vashisht 1 Department of Mathematics University of Delhi, Delhi-110007, India saakshi.garg@yahoo.com, & lalitkvashisht@gmail.com Abstract Motivated by a new concept of weaving frames in separable Hilbert spaces by Bemrose, Casazza, Gr¨ ochenig, Lammers and Lynch [Weaving Frames, Oper. Matrices, 10 (4) (2016), 1093–1116], we study weaving properties of K-fusion frames in Hilbert space. We present necessary and sufficient conditions for weaving K-fusion frames in separable Hilbert spaces. A Paley-Wiener type perturbation result for weaving K-fusion frames is given. Subject class [2010]:42C15; 42C30; 42C40. Keywords: Frames, atomic system, fusion frames, weaving frames, perturbation 1 Introduction In 1952, Duffin and Schaffer [12] introduced the concept of a frame in separable Hilbert spaces. A countable sequence {x k } k∈I ⊂H is called a frame (or Hilbert frame ) for H if there exist constants 0 <α o ≤ β o < ∞ such that α o ‖x‖ 2 ≤  k∈I |〈x, x k 〉| 2 ≤ β o ‖x‖ 2 for all x ∈H. The numbers α o and β o are called lower and upper frame bounds, respectively. If it is possible to choose α o = β o , then we say that the frame {x k } k∈I is tight. Following three operators are associated with a frame {x k } ∞ k=1 for H: pre-frame operator T : ℓ 2 (I) →H, T {c k } ∞ k=1 =  k∈I c k x k , {c k } k∈I ∈ ℓ 2 (I), analysis operator (adjoint of T ) T ∗ : H→ ℓ 2 (I), T ∗ x = {〈f,x k 〉} k∈I , x ∈H, frame operator Λ= TT ∗ : H→H, Λx =  k∈I 〈x, x k 〉x k ,x ∈H. 1 The second author is supported by R & D Doctoral Research Programme, University of Delhi, Delhi- 110007, India (Grant No.: RC/2015/9677).