Document Type : Research Paper

**Authors**

Department of Mathematics, Faculty of Science, University of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran.

**Abstract**

In this paper, we give some conditions under which the finite sum of continuous $g$-frames is again a continuous $g$-frame. We give necessary and sufficient conditions for the continuous $g$-frames $\Lambda=\left\{\Lambda_w \in B\left(H,K_w\right): w\in \Omega\right\}$ and $\Gamma=\left\{\Gamma_w \in B\left(H,K_w\right): w\in \Omega\right\}$ and operators $U$ and $V$ on $H$ such that $\Lambda U+\Gamma V=\{\Lambda_w U+\Gamma_w V \in B\left(H,K_w\right): w\in \Omega\}$ is again a continuous $g$-frame. Moreover, we obtain some sufficient conditions under which the finite sum of continuous $g$-frames are stable under small perturbations.

**Keywords**

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