Controlled K−g−Fusion Frames in Hilbert C∗−Modules

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Mohamed Rossafi
Fakhr-dine Nhari

Abstract

Controlled frames have been the subject of interest because of its ability to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we introduce the concepts of controlled g−fusion frame and controlled K−g−fusion frame in Hilbert C∗−modules and we give some properties. Also, we study the perturbation problem of controlled K−g−fusion frame. Moreover, an illustrative example is presented to support the obtained results.

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References

  1. A. Alijani, M. Dehghan, ∗-frames in Hilbert C ∗ -modules, U.P.B. Sci. Bull. Ser. A, 73 (4) (2011), 89–106.
  2. L. Arambasic, On frames for countably generated Hilbert C∗-modules, Proc. Amer. Math. Soc. 135 (2006), 469–478. https://doi.org/10.1090/S0002-9939-06-08498-X.
  3. P. Balazs, J.-P. Antoine, A. Grybos, Weighted and controlled frames: mutual relationship and first numerical properties, Int. J. Wavelets Multiresolut. Inf. Process. 08 (2010), 109–132. https://doi.org/10.1142/S0219691310003377.
  4. R.J. Duffin, 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.
  5. M. Frank, D. R. Larson, A-module frame concept for Hilbert C ∗ -modules, functional and harmonic analysis of wavelets, Contempt. Math. 247 (2000), 207-233.
  6. D. Gabor, Theory of communication, J. Inst. Electric. Eng. Part III, 93 (1946), 429–457.
  7. S. Kabbaj, M. Rossafi, ∗-operator Frame for End ∗ A(H), Wavelet Linear Algebra, 5 (2018), 1-13.
  8. I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), 839. https://doi.org/10.2307/2372552.
  9. A. Khosravi, B. Khosravi, Fusion frames and g-frames in Hilbert C ∗ -modules, Int. J. Wavelets Multiresolut. Inf. Process. 06 (2008), 433–446. https://doi.org/10.1142/S0219691308002458.
  10. M. R. Kouchi and A. Rahimi, On controlled frames in Hilbert C ∗−modules, Int. J. Wavelets Multiresolut. Inf. Process. 15 (2017), 1750038. https://doi.org/10.1142/S0219691317500382.
  11. F. D. Nhari, R. Echarghaoui, M. Rossafi, K−g−fusion frames in Hilbert C ∗−modules, Int. J. Anal. Appl. 19 (2021), 836-857. https://doi.org/10.28924/2291-8639-19-2021-836.
  12. W. Paschke, Inner product modules over B ∗ -algebras, Trans. Amer. Math. Soc. 182 (1973), 443-468. https://doi.org/10.1090/S0002-9947-1973-0355613-0.
  13. M. Rossafi, S. Kabbaj, ∗-K-operator frame for End ∗ A(H), Asian-Eur. J. Math. 13 (2020), 2050060. https://doi.org/10.1142/S1793557120500606.
  14. M. Rossafi, S. Kabbaj, Operator frame for End ∗ A(H), J. Linear Topol. Algebra, 8 (2019), 85-95.
  15. M. Rossafi, S. Kabbaj, ∗-K-g-frames in Hilbert A-modules, J. Linear Topol. Algebra, 7 (2018), 63-71.
  16. M. Rossafi, S. Kabbaj, ∗-g-frames in tensor products of Hilbert C ∗ -modules, Ann. Univ. Paedagog. Crac. Stud. Math. 17 (2018), 17-25.
  17. M. Rossafi, S. Kabbaj, Generalized frames for B(H, K), Iran. J. Math. Sci. Inf. accepted.
  18. X. Fang, M.S. Moslehian, Q. Xu, On majorization and range inclusion of operators on Hilbert C∗-modules, Linear Multilinear Algebra. 66 (2018), 2493–2500. https://doi.org/10.1080/03081087.2017.1402859.