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**

*Continuous $g$-frames in Hilbert spaces*, Southeast Asian Bull. Math., 32 (2008), pp. 1-19.

[2] S.T. Ali, J.P. Antoine, and J.P. Cazeau,

*Continuous frames in Hilbert spaces*, Ann. Physics, 222 (1993), pp. 1-37.

[3] O. Christensen,

*An Introduction to Frames and Riesz Bases*, Applied and Numerical Harmonic Analysis, Birkhauser Boston, 2016.

[4] O. Christensen,

*Frame perturbations*, Proc. Amer. Math. Soc., 123 (1995), pp. 1217-1220.

[5] R. Chugh and S. Goel,

*On finite sum of $g$-frames and near exact $g$-frames*, Electron. J. Math. Anal. Appl., 2 (2014), pp. 73-80.

[6] I. Daubechies, A. Grossmann, and Y. Mayer,

*Painless nonorthogonal expansions*, J. Math. Phys., 27 (1986), pp. 1271-128.

[7] R.J. Duffin and A.C. Schaeffer,

*A class of nonharmonic Fourier series*, Trans. Amer. Math. Soc., 72 (1952), pp. 341-366.

[8] G. Kaiser,

*A Friendly Guide to Wavelets*, Birkhauser Boston, MA, 1995.

[9] D. Li, J. Leng, T. Huang, and G. Sun,

*On sum and stability of $g$-frames in Hilbert spaces*, Linear Multilinear Algebra, 66 (2018), pp. 1578-1592.

[10] M. Madadian and M. Rahmani,

*$G$-frame sequence oprtators, $cg$-Riesz bases and sum of $cg$-frames*, Int. Math. Forum, 68 (2011), 3357-3369.

[11] G.J. Murphy,

*C$^*$-Algebras and Operator Theory*, Academic Press, San Diego, California, 1990.

[12] S. Obeidat, S. Samarah, P.G. Casazza, and J.C. Tremain,

*Sums of Hilbert space frames*, J. Math. Anal. Appl., 351 (2009), pp. 579-585.

[13] W. Sun,

*$G$-frames and $g$-Riesz bases*, J. Math. Anal. Appl., 322 (2006), pp. 437-452.

[14] W. Sun,

*Stability of $g$-frames*, J. Math. Anal. Appl., 326 (2007), pp. 858-868.