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

TL;DR

This paper extends Douglas' factorization theorem to Hilbert pro-$C^{\

Contribution

It introduces generalized inverse operators, defines atomic systems and K-frames in Hilbert pro-$C^{\

Findings

Established Douglas' factorization theorem type for Hilbert pro-$C^{\

Demonstrated properties of K-frames using the theorem

Proved the sum of two K-frames under certain conditions is again a K-frame

Abstract

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.