Document Type: Research Paper

**Authors**

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

**Abstract**

In this paper, we introduce $(\mathcal{C},\mathcal{C}')$-controlled continuous $g$-Bessel families and their multipliers in Hilbert spaces and investigate some of their properties. We show that under some conditions sum of two $(\mathcal{C},\mathcal{C}')$-controlled continuous $g$-frames is a $(\mathcal{C},\mathcal{C}')$-controlled continuous $g$-frame. Also, we investigate when a $(\mathcal{C},\mathcal{C}')$-controlled continuous $g$-Bessel multiplier is a p-Schatten class operator.

**Keywords**

- Controlled continuous $g$-frames
- $(mathcal{C}
- mathcal{C}')$-controlled continuous $g$-Bessel families
- Multiplier of continuous $g$-frames

**Main Subjects**

[1] M.R. Abdollahpour and Y. Alizadeh, *Multipliers of Continuous $G$-Frames in Hilbert spaces*, Bull. Iranian. Math. Soc., 43 (2017), pp. 291-305.

[2] M.R. Abdollahpour and M.H. Faroughi, *Continous g-Frames in Hilbert spaces*, Southeast asian Bulletin of Mathematics, 32 (2008), pp. 1-19.

[3] P. Balazs, *Basic definition and properties of Bessel multipliers*, J. Math. Anal. Appl., 325 (2007), pp. 571-585.

[4] P. Balazs, J.P. Antoine, and A. Grybos, *Weighted and controlled frames*, Int. J. Wavelets Multiresolut Inf. Prosses., 8 (2010), pp. 109-132.

[5] P. Balazs, D. Bayer, and A. Rahimi, *Multipliers for continuous frames in Hilbert spaces*, J. Phys. A: Math. Theory., 45 (2012), pp. 1-20.

[6] I. Bogdanova, P. Vandergheynst, J.P. Antoine, L. Jacques, and M. Morvidone, *Stereographic wavelet frames on sphere*, Applied Comput. Harmon. Anal., 19 (2005), pp. 223-252.

[7] O. Christensen, *An Introduction to Frames and Riesz Bases*, Birkhauser Boston, 2003.

[8] R.J. Duffin and A.C. Schaeffer, *A class of nonharmonic Fourier seris*, Trans. Amer. Math. Soc., 72 (1952), pp. 341-366.

[9] L.O. Jacques, *Reperes et couronne solaire*, These de Doctorat, Univ. Cath. Louvain, Louvain-la-Neuve. 2004.

[10] G.J. Murphy, *$C^*$-algebras and operator theory*, Academic Press Inc., 1990.

[11] A. Rahimi and A. Fereydooni, *Controlled $G$-Frames and Their $G$-Multipliers in Hilbert spaces*, An. St. Univ. Ovidius Constanta, versita., 21 (2013), pp. 223-236.

[12] W. Sun, *G-frames and g-Riesz bases*, J. Math. Anal. Appl., 322 (2006), pp. 437-452.