Vali-e-Asr university of Rafsanjan Wavelet and Linear Algebra 2383-1936 13 1 2026 07 01 Results on‏ ‎operator relationships and their ranges equality in \( C^* \)-modules 47 56 737858 10.22072/wala.2026.2080618.1483 EN Mahdi Ali-Akbari Department of Computer Engineering, University of Torbat Heydarieh, Torbat Heydarieh, Iran Mehdi Mohammadzadeh Karizaki Department of Computer Engineering, University of Torbat Heydarieh, Torbat Heydarieh, Iran 0000-0001-7645-1076 Journal Article 2025 12 12 In this paper, by employing the concept of operator ranges and their equality, and utilizing the invertibility of certain operators, we establish the equality of ranges of related operators in Hilbert <br>\‎(C^{*}\)-modules. The concept of operator ranges, combined with the invertibility of these operators, provides a tool for identifying and proving range equalities of related operators.‎ &lrm Range Equality&lrm , Moore&ndash Penrose inverse, Orthogonal Projection&lrm , Hilbert $C^*$-module https://wala.vru.ac.ir/article_737858_245493d4e9352cf3954b15233decaca6.pdf