Sum of $g-$frames in Hilbert $C^{\ast}-$modules
Abdellatif Lfounoune, Hafida Massit, Abdelilah Karara, Mohamed Rossafi
TL;DR
The paper addresses when sums of $g$-frames in Hilbert $C^*$-modules retain the $g$-frame property, showing that the key is the lower-boundedness (positivity) of the frame operator $S$ via $S \ge \alpha I$ for some $\alpha>0$. It develops a framework linking the surjectivity of the synthesis operator to $S>0$ and provides constructive, sufficient conditions for the sum of $g$-frames (or a $g$-frame and a $g$-Bessel sequence) to remain a $g$-frame, including invertibility/positivity constraints on related operators and cross-terms. The work includes concrete C*-algebraic examples (matrix algebras and $C([0,1])$) showing that without these conditions, sums can fail to be $g$-frames, thus highlighting the unifying role of operator invertibility in the stability of $g$-frames. Overall, the results extend existing theorems on g-frames, providing a cohesive treatment of sums, perturbations, and stability within Hilbert $C^*$-modules and illustrating the theory with explicit algebraic settings.
Abstract
In this article, we study g-frames in Hilbert $C^*$-modules and investigate conditions under which the sum of two g-frames (or a g-frame and a g-Bessel sequence) remains a g-frame. We also address the stability of g-frames under certain perturbations and provide illustrative examples in the context of $C^*$-algebras. Our results unify and extend many of the existing theorems on g-frames, focusing on the invertibility of associated operators as a key condition for guaranteeing that sums of g-frames preserve the g-frame property.