Research PaperNo Access

Characterizations of (near) exact $g$ $g$ -frames, $g$ $g$ -Riesz bases, and Besselian $g$ $g$ -frames

Xiangchun Xiao

Department of Mathematics, Xiamen University of Technology, Xiamen 361024, P. R. China

E-mail Address: xxc570@163.com

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Yucan Zhu

Department of Mathematics and Computer Science, Fuzhou University, Fuzhou 350002, P. R. China

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, and

Guorong Zhou

Department of Mathematics, Xiamen University of Technology, Xiamen 361024, P. R. China

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https://doi.org/10.1142/S0219691319500401Cited by:4 (Source: Crossref)

Abstract

In this paper, we use a new general sequence corresponding to a $g$ $g$ -Bessel sequence ${Λ_{j} : j \in J}$ ${Λ_{j} : j \in J}$ to characterize that ${Λ_{j} : j \in J}$ ${Λ_{j} : j \in J}$ are $l^{2} ({𝒱_{j}}_{j \in J})$ -linear independent, $g$ -complete and a $g$ -frame. We also use ${Λ_{j} : j \in J}$ and the refinement ${Γ_{i j} Λ_{i} : i \in I, j \in J_{i}}$ to characterize each other to be (near) exact $g$ -frames or $g$ -Riesz bases. Finally, we give several constructions and an equivalent characterization of Besselian $g$ -frames and near exact $g$ -frames.

Keywords:

AMSC: 42C15, 42C40

References

1. M. R. Abdollahpour and A. Najati, Besselian $g$ -frames and near $g$ -Riesz bases, Appl. Anal. Discrete Math. 5 (2011) 259–270. Web of Science, Google Scholar
2. P. G. Casazza and G. Kutyniok, Finite Frames: Theory and Applications (Birkhäuser, Boston, 2012). Google Scholar
3. P. G. Casazza, G. Kutyniok and S. Li, Fusion frames and distributed processing, Appl. Comput. Harmon. Anal. 25 (2008) 114–132. Web of Science, Google Scholar
4. O. Christensen, An Introduction to Frames and Riesz Bases, 2nd edn. (Birkhäuser, Boston, 2015). Google Scholar
5. M. L. Ding and Y. C. Zhu, G-Besselian Frames in Hilbert Spaces, Acta Math. Sin. (Engl. Ser.) 26(11) (2010) 2117–2130. Web of Science, Google Scholar
6. X. X. Guo, Characterizations of disjointness of $g$ -frames and constructions of $g$ -frames in Hilbert spaces, Complex Anal. Oper. Theory 8 (2014) 1547–1563. Web of Science, Google Scholar
7. X. X. Guo, Operator parameterizations of $g$ -frames, Taiwanese J. Math. 18(1) (2014) 313–328. Web of Science, Google Scholar
8. X. X. Guo, New characterizations of $g$ -Bessel sequences and $g$ -Riesz bases in Hilbert spaces, Results Math. 68 (2015) 361–374. Web of Science, Google Scholar
9. J. R. Holub, Pre-frame operators, Besselian frame, and near-Riesz bases in Hilbert spaces, Proc. Amer. Math. Soc. 122 (1994) 79–785. Web of Science, Google Scholar
10. A. Khosravi and M. Mirzaee Azandaryani, Approximate duality of $g$ -frames in Hilbert spaces, Acta Math. Sci. 34B(3) (2014) 639–652. Web of Science, Google Scholar
11. A. Khosravi and K. Musazadeh, Fusion frames and $g$ -frames, J. Math. Anal. Appl. 342 (2008) 1068–1083. Web of Science, Google Scholar
12. J. Z. Li and Y. C. Zhu, Exact $g$ -frames in Hilbert spaces, J. Math. Anal. Appl. 374(1) (2011) 201–209. Web of Science, Google Scholar
13. W. Sun, G-frames and $g$ -Riesz bases, J. Math. Anal. Appl. 322 (2006) 437–452. Web of Science, Google Scholar
14. W. Sun, Stability of $g$ -frames, J. Math. Anal. Appl. 326 (2007) 858–868. Web of Science, Google Scholar
15. X. C. Xiao and Y. C. Zhu, Exact $K$ - $g$ -frames in Hilbert spaces, Results Math. 72(3) (2017) 1329–1339. Web of Science, Google Scholar
16. X. C. Xiao et al., G-frames with bounded linear operators, Rocky Mountain J. Math. 45(2) (2015) 675–693. Web of Science, Google Scholar
17. X. C. Xiao, Y. C. Zhu and M. L. Ding, Erasures and equalities for fusion frames in Hilbert spaces, Bull. Malays. Math. Sci. Soc. 38 (2015) 1035–1045. Web of Science, Google Scholar
18. X. C. Xiao, Y. C. Zhu and X. M. Zeng, Oblique dual frames in finite-dimensional Hilbert spaces, Int. J. Wavelets, Multiresolut. Inform. Process. 11(2) (2013) 1350011-1–1350011-14. Link, Web of Science, Google Scholar
19. Y. C. Zhu, Characterizations of $g$ -frames and $g$ -Riesz bases in Hilbert spaces, Acta Math. Sin. (Engl. Ser.) 24(10) (2008) 1727–1736. Web of Science, Google Scholar