^{}Department of Mathematics, Faculty of Science, Vali-e-Asr University of Rafsanjan, P.O. Box 7719758457, Rafsanjan, Iran.

Abstract

Generalized frames are an extension of frames in Hilbert spaces and Hilbert $C^*$-modules. In this paper, the concept ''Similar" for modular $g$-frames is introduced and all of operator duals (ordinary duals) of similar $g$-frames with respect to each other are characterized. Also, an operator dual of a given $g$-frame is studied where $g$-frame is constructed by a primary $g$-frame and an orthogonal projection. Moreover, a $g$-frame is obtained by two the $g$-frames and its operator duals are investigated. Finally, the dilation of $g$-frames is studied.

[1] A. Alijani, Generalized frames with C^{*}-valued bounds and their operator duals, Filomat, 29 (7) (2015), 1469-1479.

[2] A. Alijani and M.A. Dehghan, $G$-frames and their duals for Hilbert C^{*}-modules, Bull. Iranian Math. Soc., 38 (3) (2012), 567-580.

[3] P.G. Casazza, The art of frame theory, Taiw. J. Math., 4 (2) (2000), 129-201.

[4] P.G. Casazza and G. Kutyniok, Finite Frames: Theory and Applications, Brikhauser Basel, 2013.

[5] M.A. Dehghan and M.A. Hasankhani Fard, G-Dual frames in Hilbert spaces, U.P.B. Sci. Bull., Series A, 75 (1) (2013), 129-140.

[6] M. Frank and D.R. Larson, Frames in Hilbert C^{*}-modules and C^{*}-algebra, J. Operator theory, 48 (2002), 273-314.

[7] D. Han, W. Jing, D. Larson, and R. Mohapatra, Riesz bases and their dual modular frames in Hilbert C^{*}-modules, J. Math. Anal. Appl., 343 (2008), 246-256.

[8] A. Khosravi and B. Khosravi, Fusion frames and $g$-frames in Hilbert C^{*}-modules, Int. J. Wavelets Multiresolut. Inf. Process., 6 (2008), 433-446.

[9] A. Khosravi and B. Khosravi, $g$-frames and modular Riesz bases in Hilbert C^{*}-modules, Int. J. Wavelets Multiresolut. Inf. Process., 10 (2) (2012), 1250013/1-1250013/12.

[10] E.C. Lance, Hilbert C^{*}-modules, A Toolkit for Operator Algebraists, University of Leeds, Cambridge University Press, 1995.

[11] A. Najati and A. Rahimi, Generalized frames in Hilbert spaces, Bull. Iran Math. Soc., 35 (1) (2009), 97-109.

[12] W. Sun, $G$-Frames and $g$-Riesz bases, J. Math. Anal. Appl., 322 (2006) 437-452.