Douglas' Factorization Theorem and Atomic System in Hilbert Pro-$C^{*}$-Modules - Mohamed Rossafi; Roumaissae Eljazzar; Ram Mohapatra (Sahand Communications in Mathematical Analysis)
In the present paper, we introduce the generalized inverse operators, which have an exciting role in operator theory. We establish Douglas' factorization theorem type for the Hilbert pro-$C^{*}$-module.We introduce the notion of atomic system and $K$-frame in the Hilbert pro-$C^{*}$-module and study their relationship. We also demonstrate some properties of the $K$-frame by using Douglas' factorization theorem.Finally we demonstrate that the sum of two $K$-frames in a Hilbert pro-$C^{*}$-module with certain conditions is once again a $K$-frame.