On perturbation of Hilbert-Schmidt frames

TL;DR

This paper investigates how Hilbert-Schmidt frames behave under structured modifications, providing explicit criteria for their stability when elements are finitely or infinitely replaced.

Contribution

It introduces new explicit criteria for the stability of Hilbert-Schmidt frames under structured perturbations, including finite and infinite cases.

Findings

Finite perturbations preserve frames with bounds depending on perturbation size.

Infinite perturbations maintain frames under globally controlled conditions.

Examples demonstrate the applicability of the stability criteria.

Abstract

In this paper, we study perturbation of Hilbert-Schmidt frames under structured modifications, where the perturbation takes the form of replacing finitely or infinitely many frame elements. We establish explicit criteria under which the perturbed sequence retains the Hilbert-Schmidt frame property. In the finite case, the stability bounds depend quantitatively on the perturbation size and the number of altered elements. For the infinite case, we identify sufficient conditions ensuring stability under globally controlled perturbations. Our study includes illustrative examples demonstrating the applicability of the results.