Azam Yousefzadeheyni - Department of Mathematics, Faculty of Science, University of Mohaghegh Ardabili, ۵۶۱۹۹-۱۱۳۶۷, Ardabil, Iran.
Mohammad Reza Abdollahpour - Department of Mathematics, Faculty of Science, University of Mohaghegh Ardabili, ۵۶۱۹۹-۱۱۳۶۷, Ardabil, Iran.

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 $Lambda=left{Lambda_w in Bleft(H,K_wright): win Omegaright}$ and $Gamma=left{Gamma_w in Bleft(H,K_wright): win Omegaright}$ and operators $U$ and $V$ on $H$ such that $Lambda U+Gamma V={Lambda_w U+Gamma_w V in Bleft(H,K_wright): win Omega}$ 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.

Continuous $g$-frame, Parseval continuous $g$-frame, Continuous $g$-Bessel family, stability