Invariances of the operator properties of frame multipliers under perturbations of frames and symbol

TL;DR

This paper investigates the stability of operator properties of frame multipliers under perturbations of frames and symbols, with a focus on invertible multipliers, advancing understanding relevant to Gabor multipliers.

Contribution

It provides new results on the invariance of frame multiplier properties under perturbations, including invertibility, with implications for Gabor multiplier analysis.

Findings

Operator properties are stable under perturbations of frames and symbols.

Invertible frame multipliers retain their properties under perturbations.

Results facilitate new insights into Gabor multipliers.

Abstract

Let $\Phi$ and $\Psi$ be frames for $\cal H$ and let $M_{m,\Phi,\Psi}$ be a frame multiplier with the symbol $m$. In this paper, we restrict our investigation to show that the operator properties of $M_{m,\Phi,\Psi}$ are stable under the perturbations of $\Phi$, $\Psi$ and $m$. Also, special attention is devoted to the study of invertible frame multipliers. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Gabor multipliers which have been studied mostly by Hans Georg Feichtinger and his coauthors in recent years.