Document Type : Research Paper


1 Department of Mathematics, Faculty of Mathematics Science and Statistics, University of Birjand, Birjand 9717851367, Iran.

2 Department of Basic sciences, Birjand University of Technology, Birjand 9719866981, Iran.


In this paper, number of properties, involving invertibility, existence of Moore-Penrose inverse and etc for modular operators with the same ranges on Hilbert $C^*$-modules  are presented. Natural decompositions of operators with closed range enable us to find some relations of the product of operators with  Moore-Penrose inverses under the condition that they have  the same ranges  in Hilbert $C^*$-modules. The triple reverse order law and the mixed reverse order law in the special cases are also given. Moreover, the range property and Moore-Penrose inverse are illustrated.


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