Continuous K-Biframes in Hilbert Spaces

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-135
Published: Apr 30, 2026
Cite as:
Abdelilah Karara, Maryam Gharamah Alshehri, Mohamed Rossaf, Continuous K-Biframes in Hilbert Spaces, Int. J. Anal. Appl., 24 (2026), 135.

Main Article Content

Abdelilah Karara, Maryam Gharamah Alshehri, Mohamed Rossaf

Abstract

In this paper, we will introduce the concept of a continuous K-biframe for Hilbert spaces, and we will present various examples of continuous K-biframes. Furthermore, we investigate their characteristics from the perspective of operator theory by establishing various properties. These results are interesting and more general than what exists in the literature.

Article Details

References

  1. I. Daubechies, A. Grossmann, Y. Meyer, Painless Nonorthogonal Expansions, J. Math. Phys. 27 (1986), 1271–1283. https://doi.org/10.1063/1.527388.
  2. R.J. Duffin, A.C. Schaeffer, A Class of Nonharmonic Fourier Series, Trans. Am. Math. Soc. 72 (1952), 341–366. https://doi.org/10.2307/1990760.
  3. R.G. Douglas, On Majorization, Factorization, and Range Inclusion of Operators on Hilbert Space, Proc. Am. Math. Soc. 17 (1966), 413–415. https://doi.org/10.1090/s0002-9939-1966-0203464-1.
  4. R. El Jazzar, R. Mohamed, On Frames in Hilbert Modules Over Locally $C^{ast}$-Algebras, Int. J. Anal. Appl. 21 (2023), 130. https://doi.org/10.28924/2291-8639-21-2023-130.
  5. A. Fereydooni, A. Safapour, Pair Frames, Results Math. 66 (2014), 247–263.
  6. D. Gabor, Theory of Communications, J. Inst. Electr. Eng. 93 (1946) 429–457.
  7. A. Karara, M. Rossafi, M. Klilou, S. Kabbaj, Construction of Continuous K-g-Frames in Hilbert $C^{ast}$-Modules, Palest. J. Math. 13 (2024), 198–209.
  8. A. Karara, M. Rossafi, A. Touri, K-Biframes in Hilbert Spaces, J. Anal. 33 (2024), 235–251. https://doi.org/10.1007/s41478-024-00831-3.
  9. A. Lfounoune, A. Karara, M. Rossafi, Continuous Biframes in Hilbert $C^{ast}$-Modules, Int. J. Anal. Appl. 23 (2025), 104. https://doi.org/10.28924/2291-8639-23-2025-104.
  10. A. Lfounoune, R. El Jazzar, K-Frames in Super Hilbert $C^{ast}-$Modules, Int. J. Anal. Appl., 23 (2025), 19. https://doi.org/10.28924/2291-8639-23-2025-19.
  11. B.V. Limaye, Functional Analysis, New Age International Publishers Limited, New Delhi, (1996).
  12. H. Massit, M. Rossafi, C. Park, Some Relations Between Continuous Generalized Frames, Afr. Mat. 35 (2023), 12. https://doi.org/10.1007/s13370-023-01157-2.
  13. H. Massit, R. Eljazzar, M. Rossafi, Continuous Biframes in Hilbert Spaces, Afr. Mat. 37 (2026), 58. https://doi.org/10.1007/s13370-026-01453-7.
  14. F.D. Nhari, R. Echarghaoui, M. Rossafi, K-G-Fusion Frames in Hilbert $C^{ast}$-Modules, Int. J. Anal. Appl. 19 (2021), 836–857. https://doi.org/10.28924/2291-8639-19-2021-836.
  15. F.D. Nhari, K. Mabrouk, M. Rossafi, On Woven K-g-Frame in Hilbert $C^{*}$-Modules, J. Anal. (2026). https://doi.org/10.1007/s41478-026-01064-2.
  16. E.H. Ouahidi, M. Rossafi, Woven Continuous Generalized Frames in Hilbert $C^{ast}$-Modules, Int. J. Anal. Appl. 23 (2025), 84. https://doi.org/10.28924/2291-8639-23-2025-84.
  17. M.F. Parizi, A. Alijani, M.A. Dehghan, Biframes and Some of Their Properties, J. Inequal. Appl. 2022 (2022), 104. https://doi.org/10.1186/s13660-022-02844-7.
  18. A. Rahimi, A. Najati, Y.N. Dehghan, Continuous Frames in Hilbert Spaces, Methods Funct. Anal. Topol. 12 (2006), 170–182.
  19. M. Rossafi, F.D. Nhari, Controlled K-g-Fusion Frames in Hilbert $C^{ast}$-Modules, Int. J. Anal. Appl. 20 (2022), 1. https://doi.org/10.28924/2291-8639-20-2022-1.
  20. M. Rossafi, F.D. Nhari, K-G-Duals in Hilbert $C^{ast}$-Modules, Int. J. Anal. Appl. 20 (2022), 24. https://doi.org//10.28924/2291-8639-20-2022-24.
  21. M. Rossafi, F.D. Nhari, C. Park, S. Kabbaj, Continuous g-Frames with $C^{*}$-Valued Bounds and Their Properties, Complex Anal. Oper. Theory 16 (2022), 44. https://doi.org/10.1007/s11785-022-01229-4.
  22. M. Rossafi, S. Kabbaj, Generalized Frames for $B(H, K)$, Iran. J. Math. Sci. Inform. 17 (2022), 1–9. https://doi.org/10.52547/ijmsi.17.1.1.
  23. M. Rossafi, S. Kabbaj, $ast$-K-Operator Frame for $operatorname{End}_{mathcal{A}}^{ast}(mathcal{H})$, Asian-Eur. J. Math. 13 (2020), 2050060. https://doi.org/10.1142/S1793557120500606.
  24. M. Rossafi, S. Kabbaj, Operator Frame for $operatorname{End}_{mathcal{A}}^{ast}(mathcal{H})$, J. Linear Topol. Algebra 8 (2019), 85–95.
  25. M. Rossafi, S. Kabbaj, $ast$-K-$g$-Frames in Hilbert $mathcal{A}$-Modules, J. Linear Topol. Algebra, 7 (2018), 63–71.
  26. M. Rossafi, S. Kabbaj, $ast$-$g$-Frames in Tensor Products of Hilbert $C^{ast}$-Modules, Ann. Univ. Paedagog. Crac. Stud. Math. 17 (2018), 17–25.
  27. M. Rossafi, A. Karara, R. El Jazzar, Biframes in Hilbert $C^*$-Modules, Montes Taurus J. Pure Appl. Math. 7 (2025), 69–80.
  28. M. Rossafi, S. Kabbaj, Frames and Operator Frames for $B(H)$, Asia Math. 2 (2018), 19–23.
  29. S. Touaiher, R. El Jazzar, M. Rossafi, Properties and Characterizations of Controlled K-g-Fusion Frames Within Hilbert $C^*$-Modules, Int. J. Anal. Appl. 23 (2025), 111. https://doi.org/10.28924/2291-8639-23-2025-111.
  30. S. Touaiher, M. Rossafi, Construction of New Continuous K-g-Frames Within Hilbert $C^*$-Modules, Int. J. Anal. Appl. 23 (2025), 232. https://doi.org/10.28924/2291-8639-23-2025-232.