Weaving Continuous Controlled K-g-Gusion Frames in Hilbert Spaces


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Authors: P. GHOSH AND T. K. SAMANTA

DOI: 10.46793/KgJMat2601.115G

Abstract:

We introduce the notion of weaving continuous controlled K-g-fusion frame in Hilbert space. Some characterizations of weaving continuous controlled K-g-fusion frame have been presented. We extend some of the recent results of woven K-g-fusion frame and controlled K-g-fusion frame to woven continuous controlled K-g-fusion frame. Finally, a perturbation result of woven continuous controlled K-g-fusion frame has been studied.



Keywords:

Frame, g-fusion frame, continuous g-fusion frame, controlled frame, woven frame.



References:

[1]   R. Ahmadi, G. Rahimlou, V. Sadri and R. Z. Farfar, Constructions of K-g fusion frames and their duals in Hilbert spaces, Bull. Transilv. Univ. Brasov Ser. III. Math. Comput. Sci. 13(62) (2020), 17–32. https://doi.org/10.31926/but.mif.2020.12.61.1.2

[2]   S. T. Ali, J. P. Antonie and J. P. Gazeau, Continuous frames in Hilbert spaces, Annals of Physics 222 (1993), 1–37. https://doi.org/10.1006/aphy.1993.1016

[3]   E. Alizadeh and V. Sadri, On continuous weaving G-frames in Hilbert spaces, Wavelets and Linear Algebra 7(1) (2020), 23–36. https://doi.org/10.22072/wala.2020.114423.1248

[4]   E. Alizadeh, A. Rahimi, E. Osgooei and M. Rahman, Continuous K-G-fusion frames in Hilbert spaces, TWMS J. Pure Appl. Math. 11(1) (2021), 44–55.

[5]   E. Alizadeh and V. Sadri, Construction of weaving continuous g-frames for operators in Hilbert spaces, Probl. Anal. Issues Anal. 10(2) (2021), 3–17. https://doi.org/10.15393/j3.art.2021.9310

[6]   P. Balazs, J. P. Antonie and A. Grybos, Weighted and controlled frames: Mutual relationship and first numerical properties, Int. J. Wavelets Multiresolut. Inf. Process. 14(1) (2010), 109–132. https://doi.org/10.1142/S0219691310003377

[7]   T. Bemrose, P. G. Casazza, K. Grochenic, M. C. Lammers and R. G. Lynch, Weaving frames, Operators and Matrices 10(4) (2016), 1093–1116. https://doi.org/10.7153/oam-10-61

[8]   O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, 2008.

[9]   P. Casazza and G. Kutyniok, Frames of subspaces, Contemp. Math. 345 (2004), 87–114. https://doi.org/10.1090/conm/345/06242

[10]   P. Casazza and O. Christensen, Perturbation of operators and applications to frame theory, J. Fourier Anal. Appl. 3 (1997), 543–557. https://doi.org/10.1007/BF02648883

[11]   I. Daubechies, A. Grossmann and Y. Mayer, Painless nonorthogonal expansions, J. Math. Phys. 27(5) (1986), 1271–1283. https://doi.org/10.1063/1.527388

[12]    R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space. Proc. Amer. Math. Soc. 17 (1966), 413–415. https://doi.org/10.1080/03081087.2017.1402859

[13]   R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366.

[14]   M. H. Faroughi, A. Rahimi and R. Ahmadi, GC-fusion frames, Methods Funct. Anal. Topology 16(2) (2010), 112–119.

[15]   P. Gavruta, On the duality of fusion frames, J. Math. Anal. Appl. 333 (2007), 871–879. https://doi.org/10.1016/j.jmaa.2006.11.052

[16]   L. Gavruta, Frames for operator, Appl. Comput. Harmon. Anal. 32(1) (2012), 139–144. https://doi.org/10.1016/j.acha.2011.07.006

[17]   S. Garg, K. L. Vashisht and G. Verma, On weaving fusion frames for Hilbert spaces, International Conference on Sampling Theory and Applications (SampTA) (2017), 381–385. https://doi.org/10.1109/SAMPTA.2017.8024363

[18]   P. Ghosh and T. K. Samanta, Stability of dual g-fusion frame in Hilbert spaces, Methods Funct. Anal. Topology 26(3) (2020), 227–240.

[19]   P. Ghosh and T. K. Samanta, Generalized atomic subspaces for operators in Hilbert spaces, Math. Bohem. 147(2) (2022), 325–345. https://doi.org/10.21136/MB.2021.0130-20

[20]   P. Ghosh and T. K. Samanta, Generalized fusion frame in tensor product of Hilbert spaces, J. Indian Math. Soc. 89 (1–2) (2022), 58–71. https://doi.org/10.18311/jims/2022/29307

[21]   P. Ghosh and T. K. Samanta, Continuous controlled generalized fusion frames in Hilbert spaces, J. Indian Math. Soc. (to appear).

[22]   G. Kaiser, A Friendly Guide to Wavelets, Birkhauser, 1994.

[23]   A. Khosravi and K. Musazadeh, Controlled fusion frames, Methods Funct. Anal. Topology 18(3) (2012), 256–265.

[24]   D. Li, J. Leng and T. Huang, On weaving g-frames for Hilbert spaces, Complex Anal. Oper. Theory 14(33) (2020). https://doi.org/10.1007/s11785-020-00991-7

[25]   M. Mohammadrezaee, M. Rashidi-Kouchi, A. Nazari and A. Oloomi, Woven g-fusion frames in Hilbert spaces, Sahand Communications in Mathematical Analysis 18(3) (2021), 133–151. https://doi.org/10.22130/scma.2021.137940.870

[26]   M. Nouri, A. Rahimi and Sh. Najafizadeh, Controlled K-frames in Hilbert spaces, Int. J. Anal. Appl. 4(2) (2015), 39–50.

[27]   A. Rahimi and A. Fereydooni, Controlled g-frames and their g-multipliers in Hilbert spaces, An. Stiint. Univ. “Ovidius” Constanta Ser. Mat. 21(2) (2013), 223–236. https://doi.org/10.2478/auom-2013-0035

[28]   G. Rahimlou, V. Sadri and R. Ahmadi, Construction of controlled K-g-fusion frame in Hilbert spaces, UPB Scientific Bulletin, Series A 82(1) (2020).

[29]    R. Rezapour, A. Rahimi, E. Osgooei and H. Dehghan, Controlled weaving frames in Hilbert spaces, Infinite Dimensional Analysis Quantum Probability and Related Topics 22(1) (2019), Paper ID 1950003. https://doi.org/10.1142/S0219025719500036

[30]   R. Rezapour, A. Rahimi, E. Osgooei and H. Dehghan, Continuous controlled K-g-frames in Hilbert spaces, Indian J. Pure Appl. Math. 50 (2019), 863–875. https://doi.org/10.1007/s13226-019-0359-y

[31]   V. Sadri, Gh. Rahimlou, R. Ahmadi and R. Zarghami Farfar, Generalized fusion frames in Hilbert spaces, Infinite Dimensional Analysis Quantum Probability and Related Topics 23(2) (2020), Paper ID 2050015. https://doi.org/10.1142/S0219025720500150

[32]   V. Sadri, G. Rahimlou and R. Ahmadi, K-g-fusion woven in Hilbert spaces, TWMS J. Pure Appl. Math. 11(3) (2021), 947–958.

[33]   V. Sadri, R. Ahmadi and G. Rahimlou, On continuous weaving fusion frames in Hilbert spaces, Int. J. Wavelets Multiresolut. Inf. Process. 18(5) (2020), Paper ID 2050035, 17 pages. https://doi.org/10.1142/S0219691320500356

[34]   H. Shakoory, R. Ahamadi, N. Behzadi and S. Nami, (C,C)-Controlled g-fusion frames, Iran. J. Math. Sci. 18(1) (2023), 179–191. https://doi.org/10.52547/ijmsi.18.1.179

[35]   W. Sun, G-frames and G-Riesz bases, J. Math. Anal. 322(1) (2006), 437–452. https://doi.org/10.1016/j.jmaa.2005.09.039

[36]   L. K. Vashisht and Deepshikha, On continuous weaving frames, Adv. Pure Appl. Math. 8(1) (2017), 15–31. https://doi.org/10.1515/apam-2015-0077