# Positive Weight Function and Classification of g-Frames

## TL;DR

This paper investigates the structure of g-frames, g-Riesz bases, and g-orthonormal bases in Hilbert spaces influenced by positive weight functions and isometries, with applications to shift-invariant subspaces on the Heisenberg group.

## Contribution

It introduces a new class of linear maps characterized as g-frames, g-Riesz bases, and g-orthonormal bases based on positive weight functions and isometries, extending the theory and applications.

## Key findings

- Characterization of g-frames and related bases via weight functions.
- Application to shift-invariant subspaces on the Heisenberg group.
- New insights into the structure of frames in weighted Hilbert spaces.

## Abstract

Given a positive weight function and an isometry map on a Hilbert spaces $\mathcal{H}$, we study a class of linear maps which is a $g$-frame, $g$-Riesz basis and a $g$-orthonormal basis for $\mathcal{H}$ with respect to $\mathbb{C}$ in terms of the weight function. We apply our results to study the frame for shift-invariant subspaces on the Heisenberg group.