Document Type : Research Paper


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


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.


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.