Duals‎, ‎approximate duals and pseudo-duals of generalized frames in Hilbert C*-modules

Articles in Press

Document Type : Original Article

Author

Department of Mathematics, University of Qom, Qom, Iran.

10.22098/jfga.2025.16478.1144

Abstract

The present paper considers duals‎, ‎approximate duals and pseudo-duals of generalized frames in Hilbert C*-modules‎. ‎In particular‎, ‎the ones constructed by bounded operators inserted between the synthesis and analysis operators of a Bessel sequence are focused and characterized‎. ‎Moreover‎, ‎the mentioned notions for modular g-Riesz bases are studied and some of their properties are obtained.

Keywords

References

 1. L. Arambasic, On frames for countably generated Hilbert C∗-modules, Proc. Amer. Math.Soc. 135 (2007), 469-478.
2. L. Arambasic and D. Bakic, Frames and outer frames for Hilbert C∗-modules, Linear and multilinear algebra. 65 (2017), 381-431.
3. D. Bakic and T. Beric, On excesses of frames, Glasnik Matematicki. 50 (2015), 415-427.
4. O. Christensen and R.S. Laugesen, Approximate dual frames in Hilbert spaces and applications to Gabor frames, Sampl Theory Signal Image Process. 9 (2011), 77-90.
5. R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer.Math. Soc. 72 (1952), 341-366.
6. M. Frank and D.R. Larson, Frames in Hilbert C∗-modules and C∗-algebras, J. Operator Theory. 48 (2002), 273-314.
7. D. Han, W. Jing, D. Larson and R. Mohapatra, Riesz bases and their dual modular frames in Hilbert C∗-modules, J. Math. Anal. Appl. 343 (2008), 246-256.
8. M. Hasannasab, Modular Riesz bases versus Riesz bases in Hilbert C∗-modules, AsianEuropean Journal of Mathematics. 14 (2021), 2050149.
9. S.B. Heineken, P.M. Morillas, A.M. Benavente and M.I. Zakowicz, Dual fusion frames,Arch. Math. 103 (2014), 355-365.
10. A. Khosravi and B. Khosravi, Fusion frames and g-frames in Hilbert C∗-modules, Int.J. Wavelets Multiresolut. Inf. Process. 6 (2008), 433-446.
11. A. Khosravi and B. Khosravi, G-frames and modular Riesz bases, Int. J. Wavelets Multiresolut. Inf. Process. 10 (2012), 1250013-1-1250013-12.
12. A. Khosravi and M. Mirzaee Azandaryani, Approximate duality of g-frames in Hilbert spaces, Acta. Math. Sci. 34 (2014), 639-652.
13. E.C. Lance, Hilbert C∗-modules- A Toolkit for operator algebraists. London Math. Soc. Lecture Note Ser., Vol. 210, Cambridge University Press, Cambridge, England (1995).
14. M. Mirzaee Azandaryani, Approximate duals and nearly Parseval frames, Turk. J. Math.39 (2015), 515-526.
15. M. Mirzaee Azandaryani, On the approximate duality of g-frames and fusion frames, U.P. B. Sci. Bull. Ser A. 79 (2017), 83-93.
16. M. Mirzaee Azandaryani, An operator theory approach to the approximate duality of Hilbert space frames, J. Math. Anal. Appl. 489 (2020), 1-13 (124177).
17. M. Mirzaee Azandaryani, Approximate duals, Q-approximate duals and morphisms of Hilbert C∗-modules, Iran. J. Sci. Technol. Trans. Sci. 44 (2020), 1685-1694.
18. M. Mirzaee Azandaryani, Pseudo-duals of frames and modular Riesz bases in Hilbert C∗-modules, Sahand Communications in Mathematical Analysis. 21(4) (2024), 173-189.
19. M. Mirzaee Azandaryani and Z. Javadi, Pseudo-duals of continuous frames in Hilbert spaces, J. Pseudo-Differ. Oper. Appl. 13 (2022), 1-16.
20. M. Mirzaee Azandaryani and Z. Javadi, Pseudo-duals and closeness of continuous gframes in Hilbert spaces, AUT Journal of Mathematics and Computing. To appear.
21. W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl. 332 (2006), 437-452.
22. X. Xiao and X. Zeng, Some properties of g-frames in Hilbert C∗-modules, J. Math. Anal.Appl. 363 (2010), 399-408.
23. X. Xiao, G. Zhao and G. Zhou, Q-duals and Q-approximate duals of g-frames in Hilbert spaces, Numerical Functional Analysis and Optimization. 44 (2023), 510-528. 
Journal of Finsler Geometry and its Applications

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Available Online from 24 April 2025

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