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

Let $\mathrm{\Phi }$ and $\mathrm{\Psi }$ be frames for $\mathcal{H}$ and let $M_{m,\mathrm{\Phi },\mathrm{\Psi }}$ be a frame multiplier with the symbol $m$. In this paper, we restrict our investigation to show that the operator properties of $M_{m,\mathrm{\Phi },\mathrm{\Psi }}$ are stable under the perturbations of $\mathrm{\Phi }$, $\mathrm{\Psi }$ 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.