G-frame representation and Invertibility of g-Bessel Multipliers

TL;DR

This paper explores the structure and invertibility of g-frames in infinite-dimensional Hilbert spaces, demonstrating their decomposition into orthonormal bases, analyzing g-frame multipliers, and introducing controlled and weighted g-frames.

Contribution

It provides new characterizations of g-frames, including their decomposition into g-orthonormal bases and conditions for representing them as linear combinations, along with analysis of g-frame multipliers and new frame concepts.

Findings

Every g-frame can be written as a sum of three g-orthonormal bases.

A g-frame can be expressed as a linear combination of two g-orthonormal bases iff it is a g-Riesz basis.

Each g-Bessel multiplier is a Bessel multiplier and the paper investigates their inversion.

Abstract

In this paper we show that every g-frame for an \linebreak infinite dimensional Hilbert space $\mathcal{H}$ can be written as a sum of three g-orthonormal bases for $\mathcal{H}$. Also, we prove that every g-frame can be represented as a linear combination of two g-orthonormal bases if and only if it is a g-Riesz basis. Further, we show each g-Bessel multiplier is a Bessel multiplier and investigate the inversion of g-frame multipliers. Finally, we introduce the concept of controlled g-frames and weighted g-frames and show that the sequence induced by each controlled g-frame (resp. weighted g-frame) is a controlled frame (resp. weighted frame).