K-frames in Hilbert $C^*$-modules

TL;DR

This paper explores conditions for operators to preserve K-frames in Hilbert C*-modules, generalizes Douglas Theorem, and characterizes K-frame generators for unitary systems, advancing the theoretical understanding of frame theory in this context.

Contribution

It introduces new conditions for operator actions on K-frames, generalizes Douglas Theorem, and characterizes K-frame generators in Hilbert C*-modules, expanding the theoretical framework.

Findings

Operator actions can preserve K-frames under specific conditions

Sum of two K-frames can be a K-frame given certain criteria

Characterization of K-frame generators for unitary systems

Abstract

In this paper, firstly we investigate conditions under which the action of an operator on a $K$-frame, remain again a $K$-frame for Hilbert module E. We also give a generalization of Douglas Theorem and we shall use it to prove the sum of two $K$-frame under certain condition is again a $K$-frame. Finally, we characterize the $K$-frame generators for a unitary system in terms of operators.