Document Type : Research Paper

Authors

Department of Mathematics and Applications, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran.

Abstract

In this paper, we introduce the concept of $g$-dual frames for Hilbert $C^{*}$-modules, and then the properties and stability results of $g$-dual frames  are given.  A characterization of $g$-dual frames, approximately dual frames and dual frames of a given frame is established. We also give some examples to show that the characterization of $g$-dual frames for Riesz bases in Hilbert spaces is not satisfied in general Hilbert $C^*$-modules.

Keywords

Main Subjects

###### ##### References
[1] P. Balazs and D.T. Stoeva, Representation of the inverse of a frame multiplier, J. Math. Anal. Appl., 422 (2015), pp. 981-994.

[2] O. Christensen and R.S. Laugesen, Approximately dual frame pairs in Hilbert spaces and applications to Gabor frames, Sampl. Theory Signal Image Process. 9 (2010), pp. 77-89.

[3] M.A. Dehghan and M.A. Hasankhani Fard, $G$-dual frames in Hilbert spaces, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 75 (2013), pp. 129-140.

[4] L. Dengfeng and L. Yanting, $G$-dual frames for generalized frames, Adv. Math., (China), 45 (2016), pp. 919-931.

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

[6] M. Frank and D.R. Larson, Frames in Hilbert $C^*$-modules and $C^*$-algebras, J. Operator Theory, 48 (2002), pp. 273-314.

[7] M. Frank and D.R. Larson, A module frame concept for Hilbert $C^*$-modules, The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999), 207--233, Contemp. Math., 247, Amer. Math. Soc., Providence, RI, 1999.

[8] F. Ghobadzadeh, A. Najati, G.A. Anastassiou, and C. Park, Woven frames in Hilbert $C^*$-module spaces, J. Comput. Anal. Appl., 25 (2018), pp. 1220-1232.

[9] F. Ghobadzadeh, A. Najati, and E. Osgooei, Modular frames and invertibility of multipliers in Hilbert $C^*$-modules, (submitted).

[10] D. Han, D. Larson, W. Jing, and R.N. Mohapatra, Riesz bases and their dual modular frames in Hilbert $C^*$-modules, J. Math. Anal. Appl., 343 (2008), pp. 246-256.

[11] M.A. Hasankhanifard and M.A. Dehghan, $G$-dual function-valued frames in $L^2(0,$∞$)$, Wavel. Linear Algebra, 2 (2015), pp. 39-47.

[12] H. Javanshiri, Some properties of approximately dual frames in Hilbert spaces, Results Math., 70 (2016), pp. 475-485.

[13] E.C. Lance, Hilbert $C^*$-modules - a toolkit for operator algebraists, London Mathematical Society Lecture Note Series, vol. 210. Cambridge University Press, England, 1995.

[14] H. Li, A Hilbert $C^*$-module admitting no frames, Bull. London Math. Soc., 42 (2010), pp. 388-394.

[15] M. Mirzaee Azandaryani, Approximate duals and nearly Parseval frames, Turkish J. Math., 39 (2015), pp. 515-526.

[16] G.J. Murphy, $C^*$-algebras and operator theory, Academic Press, San Diego, 1990.

[17] A. Najati, M. Mohammadi Saem, and P. Guavruta, Frames and operators in Hilbert $C^*$-modules, Oper. Matrices, 10 (2016), 73-81.

[18] M. Rashidi-Kouchi, A. Nazari, and M. Amini, On stability of $g$-frames and $g$-Riesz bases in Hilbert $C^*$-modules, Int. J. Wavelets Multiresolut. Inf. Process., 12 (2014), pp. 1-16.

[19] M. Rashidi-Kouchi and A. Rahimi, Controlled frames in Hilbert $C^*$-modules, Int. J. Wavelets Multiresolut. Inf. Process., 15 (2017), pp. 1-15.

[20] D.T. Stoeva and P. Balazs, Invertibility of multipliers, Appl. Comput. Harmon. Anal., 33 (2012), pp. 292-299.