Redundancy, weaving and $Q$-dual of $K$-g-frames in Hilbert spaces
Yıl 2024,
Cilt: 53 Sayı: 3, 595 - 607, 27.06.2024
Xiang Chun Xiao
Guo Ping Zhao
Guorong Zhou
Öz
In this paper we study exact $K$-g-frames, weaving of $K$-g-frames and $Q$-duals of g-frames in Hilbert spaces. We present a sufficient condition for a g-Bessel sequence to be an exact $K$-g-frame. Given two woven pairs $(\Lambda, \Gamma)$ and $(\Theta, \Delta)$ of $K$-g-frames, we investigate under what conditions $\Lambda$ can be $K$-g-woven with $\Delta$ if $\Gamma$ is $K$-g-woven with $\Theta$. Given a $K$-g-frame $\Lambda$ and its dual $\Gamma$ on $\mathcal{U}$, we construct a new pair based on $\Lambda$ and $\Gamma$ so that they are woven on a subspace $R(K)$ of $\mathcal{U}$. Finally, we characterize the $Q$-dual of g-frames using their induced sequences.
Destekleyen Kurum
Natural Science Foundation of Fujian Province, China; Xiamen University of Technology
Proje Numarası
2020J01267 and 2021J011192; 40199071 and 50419004
Kaynakça
- [1] E. Andruchow, J. Antezana and G. Corach, Topology and smooth structure for pseudoframes,
Integr. Equat. Oper. Th., 67, 451-466, 2010.
- [2] M.M. Azandaryani, On the approximate duality of g-frames and fusion frames, U.P.B.
Sci. Bull. Series A, 79 (2), 83-94, 2017.
- [3] M.M. Azandaryani, An operator theory approach to the approximate duality of Hilbert
space frames, J. Math. Anal. Appl. 489, 124177, 2020.
- [4] T. Bemrose and K. Grochenig et al. Weaving frames, Oper. Matrices, 10 (4), 1093-
1116, 2016.
- [5] P.G. Casazza and G. Kutyniok, Frames of subspaces, Contemp. Math., Amer. Math.
Soc., Providence, RI, 345, 87-113, 2004.
- [6] P.G. Casazza, G. Kutyniok and Li, S. Fusion frames and distributed processing, Appl.
Comput. Harmon. Anal. 25 (1), 114-132, 2008.
- [7] O. Christensen, An introduction to frames and Riesz bases, Second edition,
Birkhäuser, Boston, 2015.
- [8] Deepshikha and A. Samanta, On weaving generalized frames and generalized Riesz
bases, Bull. Malays. Math. Sci. Soc. 44, 3361-3375, 2021.
- [9] Deepshikha, L.K. Vashisht and G. Verma, Generalized weaving frames for operators
in Hilbert spaces, Results Math. 72 (3), 1369-1391, 2017.
- [10] L. Găvruţa, Frames for operators, Appl. Comp. Harm. Anal. 32, 139-144, 2012.
- [11] X.X. Guo, Joint similarities and parameterizations for dilations of dual g-frame pairs
in Hilbert spaces, Acta Math. Sin. ( Engl. Ser.) 35, 1827-1840, 2019.
- [12] S.B. Heineken, P.M. Morillas and A.M. Benavente, et al., Dual fusion frames, Arch.
Math., 103, 355-365, 2014.
- [13] A. Khosravi and M.M. Azandaryani, Approximate duality of g-frames in Hilbert
spaces, Acta Math. Sci. 34B (3), 639-652, 2014.
- [14] A. Khosravi and J.S. Banyarani, Weaving g-frames and weaving fusion frames, Bull.
Malays. Math. Sci. Soc. 42, 3111-3129, 2019.
- [15] J.Z. Li and Y.C. Zhu, Exact g-frames in Hilbert spaces, J. Math. Anal. Appl. 374 (1),
201-209, 2011.
- [16] E.A. Moghaddam and A.A. Arefijamaal, On excesses and duality in woven frames,
Bull. Malays. Math. Sci. Soc. 44, 3361-3375, 2021.
- [17] W.C. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl. 322, 437-452, 2006.
- [18] X.C. Xiao and Y.C. Zhu, Exact K-g-frames in Hilbert spaces, Results Math. 72 (3),
1329-1339, 2017.
- [19] X.C. Xiao, Y.C. Zhu and L. Găvruţa, Some properties of K-frames in Hilbert spaces,
Results Math. 63, 1243-1255, 2013.
- [20] X.C. Xiao, Y.C. Zhu and Z.B. Shu et al., G-frames with bounded linear operators,
Rocky Mountain J. Math. 45 (2), 675-693, 2015.
- [21] X.C. Xiao, K. Yan and G.P. Zhao et al., Tight K-frames and weaving of K-frames, J.
Pseudo-Differ. Oper. Appl. 12 (1), 1, 2021.
- [22] X.C. Xiao, G.R. Zhou and Y.C. Zhu, Weaving of K-g-frames in Hilbert spaces, ScienceAsia,
45 (3), 285-291, 2019.
- [23] Z.Q. Xiang, On K-duality and redundancy of K-g-frames, Ric. Mat., 2021.
https://doi.org/10.1007/s11587-021-00600-5
- [24] Z.Q. Xiang, Some new results of weaving K-frames in Hilbert spaces, Numer. Funct.
Anal. Optim. 42, 409-429, 2021.
- [25] Y.C. Zhu,Characterizations of g-frames and g-Riesz bases in Hilbert spaces, Acta
Math. Sin. (Engl. Ser.) 24 (10), 1727-1736, 2008.
Yıl 2024,
Cilt: 53 Sayı: 3, 595 - 607, 27.06.2024
Xiang Chun Xiao
Guo Ping Zhao
Guorong Zhou
Proje Numarası
2020J01267 and 2021J011192; 40199071 and 50419004
Kaynakça
- [1] E. Andruchow, J. Antezana and G. Corach, Topology and smooth structure for pseudoframes,
Integr. Equat. Oper. Th., 67, 451-466, 2010.
- [2] M.M. Azandaryani, On the approximate duality of g-frames and fusion frames, U.P.B.
Sci. Bull. Series A, 79 (2), 83-94, 2017.
- [3] M.M. Azandaryani, An operator theory approach to the approximate duality of Hilbert
space frames, J. Math. Anal. Appl. 489, 124177, 2020.
- [4] T. Bemrose and K. Grochenig et al. Weaving frames, Oper. Matrices, 10 (4), 1093-
1116, 2016.
- [5] P.G. Casazza and G. Kutyniok, Frames of subspaces, Contemp. Math., Amer. Math.
Soc., Providence, RI, 345, 87-113, 2004.
- [6] P.G. Casazza, G. Kutyniok and Li, S. Fusion frames and distributed processing, Appl.
Comput. Harmon. Anal. 25 (1), 114-132, 2008.
- [7] O. Christensen, An introduction to frames and Riesz bases, Second edition,
Birkhäuser, Boston, 2015.
- [8] Deepshikha and A. Samanta, On weaving generalized frames and generalized Riesz
bases, Bull. Malays. Math. Sci. Soc. 44, 3361-3375, 2021.
- [9] Deepshikha, L.K. Vashisht and G. Verma, Generalized weaving frames for operators
in Hilbert spaces, Results Math. 72 (3), 1369-1391, 2017.
- [10] L. Găvruţa, Frames for operators, Appl. Comp. Harm. Anal. 32, 139-144, 2012.
- [11] X.X. Guo, Joint similarities and parameterizations for dilations of dual g-frame pairs
in Hilbert spaces, Acta Math. Sin. ( Engl. Ser.) 35, 1827-1840, 2019.
- [12] S.B. Heineken, P.M. Morillas and A.M. Benavente, et al., Dual fusion frames, Arch.
Math., 103, 355-365, 2014.
- [13] A. Khosravi and M.M. Azandaryani, Approximate duality of g-frames in Hilbert
spaces, Acta Math. Sci. 34B (3), 639-652, 2014.
- [14] A. Khosravi and J.S. Banyarani, Weaving g-frames and weaving fusion frames, Bull.
Malays. Math. Sci. Soc. 42, 3111-3129, 2019.
- [15] J.Z. Li and Y.C. Zhu, Exact g-frames in Hilbert spaces, J. Math. Anal. Appl. 374 (1),
201-209, 2011.
- [16] E.A. Moghaddam and A.A. Arefijamaal, On excesses and duality in woven frames,
Bull. Malays. Math. Sci. Soc. 44, 3361-3375, 2021.
- [17] W.C. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl. 322, 437-452, 2006.
- [18] X.C. Xiao and Y.C. Zhu, Exact K-g-frames in Hilbert spaces, Results Math. 72 (3),
1329-1339, 2017.
- [19] X.C. Xiao, Y.C. Zhu and L. Găvruţa, Some properties of K-frames in Hilbert spaces,
Results Math. 63, 1243-1255, 2013.
- [20] X.C. Xiao, Y.C. Zhu and Z.B. Shu et al., G-frames with bounded linear operators,
Rocky Mountain J. Math. 45 (2), 675-693, 2015.
- [21] X.C. Xiao, K. Yan and G.P. Zhao et al., Tight K-frames and weaving of K-frames, J.
Pseudo-Differ. Oper. Appl. 12 (1), 1, 2021.
- [22] X.C. Xiao, G.R. Zhou and Y.C. Zhu, Weaving of K-g-frames in Hilbert spaces, ScienceAsia,
45 (3), 285-291, 2019.
- [23] Z.Q. Xiang, On K-duality and redundancy of K-g-frames, Ric. Mat., 2021.
https://doi.org/10.1007/s11587-021-00600-5
- [24] Z.Q. Xiang, Some new results of weaving K-frames in Hilbert spaces, Numer. Funct.
Anal. Optim. 42, 409-429, 2021.
- [25] Y.C. Zhu,Characterizations of g-frames and g-Riesz bases in Hilbert spaces, Acta
Math. Sin. (Engl. Ser.) 24 (10), 1727-1736, 2008.