Redundancy, weaving and $Q$-dual of $K$-g-frames in Hilbert spaces

Abstract

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.

Keywords

• K-g-frame
• exact K-g-frame
• weaving
• Q-dual

Supporting Institution

Natural Science Foundation of Fujian Province, China; Xiamen University of Technology

Project Number

2020J01267 and 2021J011192; 40199071 and 50419004

References

1. [1] E. Andruchow, J. Antezana and G. Corach, Topology and smooth structure for pseudoframes, Integr. Equat. Oper. Th., 67, 451-466, 2010.
2. [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. [3] M.M. Azandaryani, An operator theory approach to the approximate duality of Hilbert space frames, J. Math. Anal. Appl. 489, 124177, 2020.
4. [4] T. Bemrose and K. Grochenig et al. Weaving frames, Oper. Matrices, 10 (4), 1093- 1116, 2016.
5. [5] P.G. Casazza and G. Kutyniok, Frames of subspaces, Contemp. Math., Amer. Math. Soc., Providence, RI, 345, 87-113, 2004.
6. [6] P.G. Casazza, G. Kutyniok and Li, S. Fusion frames and distributed processing, Appl. Comput. Harmon. Anal. 25 (1), 114-132, 2008.
7. [7] O. Christensen, An introduction to frames and Riesz bases, Second edition, Birkhäuser, Boston, 2015.
8. [8] Deepshikha and A. Samanta, On weaving generalized frames and generalized Riesz bases, Bull. Malays. Math. Sci. Soc. 44, 3361-3375, 2021.

9. [9] Deepshikha, L.K. Vashisht and G. Verma, Generalized weaving frames for operators in Hilbert spaces, Results Math. 72 (3), 1369-1391, 2017.
10. [10] L. Găvruţa, Frames for operators, Appl. Comp. Harm. Anal. 32, 139-144, 2012.
11. [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. [12] S.B. Heineken, P.M. Morillas and A.M. Benavente, et al., Dual fusion frames, Arch. Math., 103, 355-365, 2014.
13. [13] A. Khosravi and M.M. Azandaryani, Approximate duality of g-frames in Hilbert spaces, Acta Math. Sci. 34B (3), 639-652, 2014.
14. [14] A. Khosravi and J.S. Banyarani, Weaving g-frames and weaving fusion frames, Bull. Malays. Math. Sci. Soc. 42, 3111-3129, 2019.
15. [15] J.Z. Li and Y.C. Zhu, Exact g-frames in Hilbert spaces, J. Math. Anal. Appl. 374 (1), 201-209, 2011.
16. [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. [17] W.C. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl. 322, 437-452, 2006.
18. [18] X.C. Xiao and Y.C. Zhu, Exact K-g-frames in Hilbert spaces, Results Math. 72 (3), 1329-1339, 2017.
19. [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. [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. [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. [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. [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. [24] Z.Q. Xiang, Some new results of weaving K-frames in Hilbert spaces, Numer. Funct. Anal. Optim. 42, 409-429, 2021.
25. [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.