^{}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.

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