# BESSEL MULTIPLIERS AND APPROXIMATE DUALS IN HILBERT C∗ -MODULES

• Azandaryani, Morteza Mirzaee
• Published : 2017.07.01
• 13 4

#### Abstract

Two standard Bessel sequences in a Hilbert $C^*$-module are approximately duals if the distance (with respect to the norm) between the identity operator on the Hilbert $C^*$-module and the operator constructed by the composition of the synthesis and analysis operators of these Bessel sequences is strictly less than one. In this paper, we introduce (a, m)-approximate duality using the distance between the identity operator and the operator defined by multiplying the Bessel multiplier with symbol m by an element a in the center of the $C^*$-algebra. We show that approximate duals are special cases of (a, m)-approximate duals and we generalize some of the important results obtained for approximate duals to (a, m)-approximate duals. Especially we study perturbations of (a, m)-approximate duals and (a, m)-approximate duals of modular Riesz bases.

#### Keywords

Hilbert $C^*$-module;Bessel multiplier;approximate duality;modular Riesz basis

#### References

1. L. Arambasic, On frames for countably generated Hilbert C*-modules, Proc. Amer. Math. Soc. 135 (2007), no. 2, 469-478. https://doi.org/10.1090/S0002-9939-06-08498-X
2. P. Balazs, Basic definition and properties of Bessel multipliers, J. Math. Anal. Appl. 325 (2007), no. 1, 571-585. https://doi.org/10.1016/j.jmaa.2006.02.012
3. P. Balazs, Hilbert Schmidt operators and frames classification, approximation by multipliers and algorithms, Int. J. Wavelets Multiresolut. Inf. Process. 6 (2008), no. 2, 315-330. https://doi.org/10.1142/S0219691308002379
4. P. Balazs, J. P. Antoine, and A. Grybos, Weighted and controlled frames, mutual relationship and first numerical properties, Int. J. Wavelets Multiresolut. Inf. Process. 8 (2010), no. 1, 109-132. https://doi.org/10.1142/S0219691310003377
5. P. Balazs, D. Bayer, and A. Rahimi, Multipliers for continuos frames in Hilbert spaces, J. Phys. A: Math. Theor. 45 (2012), 244023, 20 pages.
6. P. Balazs, H. G. Feichtinger, M. Hampejs, and G. Kracher, Double preconditioning for Gabor frames, IEEE Trans. Signal Process. 54 (2006), 4597-4610. https://doi.org/10.1109/TSP.2006.882100
7. P. Balazs, D. T. Stoeva, and J. P. Antoine, Classification of general sequences by frame-related operators, Sampl. Theory Signal Image Process. 10 (2011), no. 1-2, 151-170.
8. A. Bourouihiya, The tensor product of frames, Sampl. Theory Signal Image Process. 7 (2008), no. 1, 65-76.
9. H.-Q. Bui and R. S. Laugesen, Frequency-scale frames and the solution of the Mexican hat problem, Constr. Approx. 33 (2011), no. 2, 163-189. https://doi.org/10.1007/s00365-010-9098-3
10. 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.
11. I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), no. 5, 1271-1283. https://doi.org/10.1063/1.527388
12. R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341-366. https://doi.org/10.1090/S0002-9947-1952-0047179-6
13. H. G. Feichtinger and K. Grochenig, Banach spaces related to integrable group representations and their atomic decomposition I, J. Funct. Anal. 86 (1989), no. 2, 307-340. https://doi.org/10.1016/0022-1236(89)90055-4
14. H. G. Feichtinger and N. Kaiblinger, Varying the time-frequency lattice of Gabor frames, Trans. Amer. Math. Soc. 356 (2004), no. 5, 2001-2023. https://doi.org/10.1090/S0002-9947-03-03377-4
15. M. Frank and D. R. Larson, Frames in Hilbert C*-modules and C*-algebras, J. Operator Theory. 48 (2002), no. 2, 273-314.
16. J. E. Gilbert, Y. S. Han, J. A. Hogan, J. D. Lakey, D. Weiland, and G. Weiss, Smooth molecular decompositions of functions and singular integral operators, Mem. Amer. Math. Soc. 156 (2002), no. 742, 1-74.
17. 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), no. 1, 246-256. https://doi.org/10.1016/j.jmaa.2008.01.013
18. M. Holschneider, Waveletes. An analysis tool, Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995.
19. A. Khosravi and B. Khosravi, Frames and bases in tensor products of Hilbert spaces and Hilbert C*-modules, Proc. Indian Acad. Sci. Math. Sci. 117 (2007), no. 1, 1-12. https://doi.org/10.1007/s12044-007-0001-5
20. A. Khosravi and B. Khosravi, Fusion frames and g-frames in Hilbert C*-modules, Int. J. Wavelets Multiresolut. Inf. Process. 6 (2008), no. 3, 433-446. https://doi.org/10.1142/S0219691308002458
21. A. Khosravi and B. Khosravi, G-frames and modular Riesz bases, Int. J. Wavelets Multiresolut. Inf. Process. 10 (2012), no. 2, 1-12.
22. A. Khosravi and M. Mirzaee Azandaryani, Fusion frames and g-frames in tensor product and direct sum of Hilbert spaces, Appl. Anal. Discrete Math. 6 (2012),no. 2, 287-303. https://doi.org/10.2298/AADM120619014K
23. A. Khosravi and M. Mirzaee Azandaryani, Approximate duality of g-frames in Hilbert spaces, Acta. Math. Sci. B Engl. Ed. 34 (2014), no. 3, 639-652. https://doi.org/10.1016/S0252-9602(14)60036-9
24. A. Khosravi and M. Mirzaee Azandaryani, Bessel multipliers in Hilbert C*-modules, Banach. J. Math. Anal. 9 (2015), no. 3, 153-163. https://doi.org/10.15352/bjma/09-3-11
25. G. Kutyniok, K. A. Okoudjou, F. Philipp, and E. K. Tuley, Scalable frames, Linear Algebra Appl. 438 (2013), no. 5, 2225-2238. https://doi.org/10.1016/j.laa.2012.10.046
26. E. C. Lance, Hilbert C*-modules: a Toolkit for Operator Algebraists, Cambridge University Press, Cambridge, 1995.
27. M. Laura Arias and M. Pacheco, Bessel fusion multipliers, J. Math. Anal. Appl. 348 (2008), 581-588. https://doi.org/10.1016/j.jmaa.2008.07.056
28. S. Li and D. Yan, Frame fundamental sensor modeling and stability of one-sided frame perturbation, Acta Appl. Math. 107 (2009), no. 1-3, 91-103. https://doi.org/10.1007/s10440-008-9419-8
29. M. Mirzaee Azandaryani, Approximate duals and nearly Parseval frames, Turk. J. Math. 39 (2015), no. 4, 515-526. https://doi.org/10.3906/mat-1408-37
30. M. Mirzaee Azandaryani, Bessel multipliers on the tensor product of Hilbert C*-modules, Int. J. Indust. Math. 8 (2016), 9-16.
31. G. J. Murphy, C*-Algebras and Operator Theory, Academic Press, San Diego, 1990.
32. A. Rahimi, Multipliers of generalized frames in Hilbert spaces, Bull. Iranian Math. Soc. 37 (2011), no. 1, 63-80.
33. A. Rahimi and P. Balazs, Multipliers for p-Bessel sequences in Banach spaces, Integral Equations Operator Theory 68 (2010), no. 2, 193-205. https://doi.org/10.1007/s00020-010-1814-7
34. M. Speckbacher and P. Balazs, Reproducing pairs and the continuous nonstationary Gabor transform on LCA groups, J. Phys. A 48 (2015), no. 39, 395201, 16 pp.
35. D. T. Stoeva and P. Balazs, Unconditional convergence and invertibility of multipliers, arXiv: 0911.2783, 2009. https://doi.org/10.1016/j.acha.2011.11.001
36. D. T. Stoeva and P. Balazs, Invertibility of multipliers, Appl. Comput. Harmon. Anal. 33 (2012), no. 2, 292-299. https://doi.org/10.1016/j.acha.2011.11.001
37. W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl. 322 (2006), no. 1, 437-452. https://doi.org/10.1016/j.jmaa.2005.09.039
38. T. Werther, Y. C. Eldar, and N. K. Subbanna, Dual Gabor frames: theory and computational aspects, IEEE Trans. Signal Process. 53 (2005), no. 11, 4147-4158. https://doi.org/10.1109/TSP.2005.857049
39. X. Xiao and X. Zeng, Some properties of g-frames in Hilbert C*-modules, J. Math. Anal. Appl. 363 (2010), no. 2, 399-408. https://doi.org/10.1016/j.jmaa.2009.08.043