# Approximation of the Inverse Frame Operator and Stability of Hilbert$-$Schmidt Frames

## TL;DR

This paper explores the properties of Hilbert-Schmidt frames in separable Hilbert spaces, focusing on their characterization, approximation of the inverse frame operator, and stability under perturbations.

## Contribution

It introduces new characterizations of HS-frames, proposes finite-dimensional approximation methods for the inverse operator, and establishes their stability under small perturbations.

## Key findings

- HS-frames share key properties with traditional frames
- The inverse frame operator can be approximated via finite-dimensional methods
- HS-frames are stable under small perturbations

## Abstract

In this paper, we study the Hilbert$-$Schmidt frame (HS-frame) theory for separable Hilbert spaces. We first present some characterizations of HS-frames and prove that HS-frames share many important properties with frames. Then, we show how the inverse of the HS-frame operator can be approximated using finite-dimensional methods. Finally, we present a classical perturbation result and prove that HS-frames are stable under small perturbations.