# Controlled $g$-frames in Hilbert $C^*$-modules

## TL;DR

This paper extends the concept of controlled frames to g-frames within Hilbert C*-modules, providing operator theoretic characterizations, relationships, and perturbation results to enhance understanding and potential applications.

## Contribution

It introduces controlled g-frames in Hilbert C*-modules, establishing their characterizations, relationships with g-frames, and perturbation properties, advancing the theoretical framework.

## Key findings

- Equivalent conditions for controlled g-frames established
- Operator theoretic characterizations provided
- Perturbation results for controlled g-frames proved

## Abstract

To improve the numerical efficiency of iterative algorithms for inverting the frame operator, the controlled frame was introduced by Balazs et al. \cite{Balazs}, and has since been given more importance. In this paper, we introduce the concept of controlled g-frames in Hilbert $C^{*}$-modules. We establish the equivalent condition for controlled $g$-frame using operator theoretic approach. We investigate some operator theoretic characterizations of controlled $g$-frames and controlled $g$-Bessel sequences. We also established the relationship between $g$-frames and controlled $g$-frames in Hilbert $C^{*}$-modules. At the end we prove some perturbation results on controlled $g$-frames.