Document Type : Research Paper
Author
Department of Mathematics, Faculty of Sciences, Vali-e-Asr University of Rafsanjan, P.O. Box 7719758457, Rafsanjan, Iran.
Abstract
Hilbert frames theory have been extended to frames in Hilbert $C^*$-modules. The paper introduces equivalent $*$-frames and presents ordinary duals of a constructed $*$-frame by an adjointable and invertible operator. Also, some necessary and sufficient conditions are studied such that $*$-frames and ordinary duals or operator duals of another $*$-frames are equivalent under these conditions. We obtain a $*$-frame by an orthogonal projection and a given $*$-frame, characterize its duals, and give a bilateral condition for commutating frame operator of a primary $*$-frame and an orthogonal projection. At the end of paper, pre-frame operator of a dual frame is computed by pre-frame operator of a general $*$-frame and an orthogonal projection.
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[2] A. Alijani and M.A. Dehghan, $*$-Frames in Hilbert $C^*$-modules, U.P.B. Sci. Bull., Series A, 73 (2011), pp. 89-106.
[3] P.G. Casazza, The art of frame theory, Taiw. J. Math., 4 (2000), pp. 129-201.
[5] M.A. Dehghan and M.A. Hasankhani Fard, G-Dual frames in Hilbert spaces, U.P.B. Sci. Bull., Series A, 75 (2013), pp. 129-140.
[6] M. Frank and D.R. Larson, Frames in Hilbert $C^*$-modules and $C^*$-algebra, J. Operator theory, 48 (2002), pp. 273-314.
[7] E.C. Lance, Hilbert $C^*$-modules, A Toolkit for Operator Algebraists, University of Leeds, Cambridge University Press, 1995.
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