Douglas' factorization theorem and atomic system in Hilbert pro$-C^{\ast }-$module

In the present paper we introduce the generalized inverse operators which have an interesting role in operator theory. We establish Douglas’ factorization theorem type for Hilbert pro-$C^{\ast }$-module. We introduce the notion of atomic system and of $K$-frame in Hilbert pro-$C^{\ast }$-module and we study the relationship between them. We also demonstrat some properties of $K$-frame by using Douglas’ factorization theorem.Finally we demonstrate that the sum of two $K$-frames in a Hilbert pro-$C^{\ast }$-module with certain conditions is once again a $K$-frame.