Document Type: Research Paper


Young Researchers and Elite Club Kahnooj Branch, Islamic Azad University, Kerman, Iran


In this paper, we investigate duality of modular g-Riesz bases and g-Riesz bases in Hilbert C*-modules. First we give some characterization of g-Riesz bases in Hilbert C*-modules, by using properties of operator theory. Next, we characterize the duals of a given g-Riesz basis in Hilbert C*-module. In addition, we obtain sufficient and necessary condition for a dual of a g-Riesz basis to be again a g-Riesz basis. We nd a situation for a g-Riesz basis to have unique dual g-Riesz basis. Also, we show that every modular g-Riesz basis is a g-Riesz basis in Hilbert C*-module but the opposite implication is not true.

Main Subjects

[1] A. Alijan, M. A. Dehghan, g-frames and their duals for Hilbert C*-modules, Bull. Iran. Math. Soci., 38(3), (2012), 567-580.

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

[3] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.

[4] I. Daubechies, A. Grossmann,Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271-1283.

[5] R.J. Dufin, A.C. Schae er, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341-366.

[6] M. Frank, D. R. Larson, A module frame concept for Hilbert C.-modules, in: Functional and Harmonic Analysis of Wavelets, San Antonio, TX, January 1999, Contemp. Math. 247, Amer. Math. Soc., Providence, RI 207-233, 2000.

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

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

[9] D. Han, W. Jing, R. Mohapatra, Perturbation of frames and Riesz bases in Hilbert C-modules, Linear Algebra Appl. 431 (2009), 746-759.

[10] A. Khosravi, B. Khosravi, Frames and bases in tensor products of Hilbert spaces and Hilbert C-modules, Proc. Indian Acad. Sci. Math. Sci. 117 (2007), 1-12.

[11] A. Khosravi, B. Khosravi, Fusion frames and g-frames in Hilbert C-modules, Int. J. Wavelets Multiresolut. Inf. Process. 6 (2008), 433-466.

[12] A. Khosravi, B. Khosravi, g-frames and modular Riesz bases in Hilbert C-modules, Int. J. Wavelets Multiresolut.
Inf. Process. 10(2) (2012), 1250013 1-12.

[13] E.C. Lance, Hilbert C-Modules: A Toolkit for Operator Algebraists, London Math. Soc. Lecture Note Ser. 210, Cambridge Univ. Press, 1995.

[14] M. Rashidi-Kouchi, A. Nazari, M. Amini, On stability of g-frames and g-Riesz bases in Hilbert C*-modules, Int. J. Wavelets Multiresolut. Inf. Process. 12(6) (2014), 1450036 1-16.

[15] W. Sun, g-Frames and g-Riesz bases, J. Math. Anal. Appl. 322 (2006), 437-452.

[16] X.-C. Xiao, X.-M. Zeng, Some properties of g-frames in Hilbert C-modules J. Math. Anal. Appl. 363 (2010), 399-408.