Advances in Pure Mathematics, 2014, 4, 373-380 Published Online August 2014 in SciRes. http://www.scirp.org/journal/apm http://dx.doi.org/10.4236/apm.2014.48048 How to cite this paper: Singh, M. and Chugh, R. (2014) Banach Λ-Frames for Operator Spaces. Advances in Pure Mathe- matics, 4, 373-380. http://dx.doi.org/10.4236/apm.2014.48048 Banach Λ-Frames for Operator Spaces Mukesh Singh 1 , Renu Chugh 2 1 Department of Mathematics, Goverment College, Bahadurgarh, India 2 Department of Mathematics, Maharishi Dayanand University, Rohtak, India Email: mukeshmdu2010@yahoo.com , chugh.1r1@gmail.com Received 11 June 2014; revised 12 July 2014; accepted 25 July 2014 Copyright © 2014 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract The Banach frame for a Banach space  can reconstruct each vector in  by the pre-frame operator or the reconstruction operator. The Banach Λ-frame for operator spaces was introduced by Kaushik, Vashisht and Khattar [Reconstruction Property and Frames in Banach Spaces, Pales- tine Journal of Mathematics, 3(1), 2014, 11-26]. In this paper we give necessary and sufficient con- ditions for the existence of the Banach Λ-frames. A Paley-Wiener type stability theorem for Λ-Banach frames is discussed. Keywords Frames, Banach Frames, Reconstruction Property, Perturbation 1. Introduction Duffin and Schaeffer in [1] while working in nonharmonic Fourier series developed an abstract framework for the idea of time-frequency atomic decomposition by Gabor [2] and defined frames for Hilbert spaces. In 1986, Daubechies, Grossmann and Meyer [3] found new applications to wavelets and Gabor transforms in which frames played an important role. Let  be an infinite dimensional separable complex Hilbert space with inner product .,. . A system { } k f ⊂  is a frame (Hilbert) for  if there exist positive constants A and B such that { } 2 2 2 2 , , for all . k Af f f Bf f ≤ ≤ ∈   (1.1) The positive constants A and B are called the lower and upper bounds of the frame { } k f , respectively. They are not unique. The inequality (1.1) is called the frame inequality of the frame. Gröchenig in [4] generalized Hilbert frames to Banach spaces. Before the concept of Banach frames was for- malized, it appeared in the foundational work of Feichtinger and Gröchenig [5] [6] related to the atomic decom- positions. Atomic decompositions appeared in the field of applied mathematics providing many applications [7].