douglas’ factorization theorem and atomic system inhilbert pro-c∗-modules

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 kframe 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.