# An appropriate representation space for controlled G-frames

## TL;DR

This paper introduces a new representation space for controlled g-frames, defines associated synthesis and analysis operators, and explores their properties, including duals and inequalities, with implications for frame theory.

## Contribution

It proposes an appropriate representation space for controlled g-frames and analyzes the properties of synthesis and analysis operators, including duals and trace class operators.

## Key findings

- The composition of synthesis and analysis operators forms a trace class operator.
- The canonical controlled g-dual yields minimal norm expansion coefficients.
- Extended known equalities and inequalities for controlled g-frames.

## Abstract

In this paper, motivating the range of operators, we propose an appropriate representation space to introduce synthesis and analysis operators of controlled g-frames and discuss the properties of these operators. Especially, we show that the operator obtained by the composition of the synthesis and analysis operators of two controlled g-Bessel sequence is a trace class operator. Also, we define the canonical controlled g-dual and show that this dual gives rise to expand coefficients with the minimal norm. Finally, we extend some known equality and inequalities for controlled g-frames.