Controlled G-Frames and Their G-Multipliers in Hilbert spaces

TL;DR

This paper introduces controlled g-frames in Hilbert spaces, proves their equivalence to g-frames, and explores the properties of associated multiplier operators, enhancing the theoretical framework for frame analysis.

Contribution

It defines controlled g-frames, establishes their equivalence to g-frames, and introduces the concept of multipliers for this family, expanding the theoretical understanding.

Findings

Controlled g-frames are equivalent to g-frames.

Multiplier operators for controlled g-frames are introduced.

Properties of the multiplier operators are analyzed.

Abstract

Multipliers have been recently introduced by P. Balazs as operators for Bessel sequences and frames in Hilbert spaces. These are operators that combine (frame-like) analysis, a multiplication with a fixed sequence (called the symbol) and synthesis. Weighted and controlled frames have been introduced to improve the numerical efficiency of iterative algorithms for inverting the frame operator Also g-frames are the most popular generalization of frames that include almost all of the frame extensions. In this manuscript the concept of the controlled g-frames will be defined and we will show that controlled g-frames are equivalent to g-frames and so the controlled operators C and C0 can be used as preconditions in applications. Also the multiplier operator for this family of operators will be introduced and some of its properties will be shown.