Abstract
In this paper we find that a result of an equivalent characterization of tight K-g-frames obtained by Huang and Shi is incorrect, and we give a sufficient condition for a given Bessel sequence to be a tight K-frame. We also characterize the weaving of K-frames in Hilbert spaces. We give several kinds of sufficient conditions such that the type \(\{T_{1} f_{i}\}_{i\in I}\) and \(\{T_{2} g_{i}\}_{i\in I}\) are K-woven (resp. woven) on \(\mathcal {H}\) or its subspace R(K), given that \(\{f_{i}\}_{i\in I}\) and \(\{g_{i}\}_{i\in I}\) are K-frames (resp. frames) on \(\mathcal {H}\) and \(T_{1}, T_{2}\) are surjective operators on \(\mathcal {H}\). Finally we discuss that we can plus two different Bessel sequences to a K-woven pair such that the new obtained pair are K-woven on \(\mathcal {H}\).
Similar content being viewed by others
Data availibility
No data, models, or code were generated or used during the study.
References
Arabyani Neyshaburi, F., Arefijamaal, A.: Some constructions of K-frames and their duals. Rocky Mt. J. Math. 47(6), 1749–1764 (2017)
Bemrose, T., Casazza, P.G., Grochenig, K., Lammers, M.C., Lynch, R.G.: Weaving frames. Oper. Matrices 10(4), 1093–1116 (2016)
Casazza, P.G., Freeman, D., Lynch, R.G.: Weaving Schauder frames. J. Approx. Theory 211, 42–60 (2016)
Casazza, P.G., Kutyniok, G.: Finite Frames: Theory and Applications. Birkhäuser, Boston (2012)
Casazza, P. G., Lynch, R. G.: Weaving properties of Hilbert space frames. In: International Conference on Sampling Theory and Applications, pp. 110–114 (2015)
Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003)
Deepshikha, Vashisht L.K: Weaving K-frames in Hilbert spaces. Results Math. 73(2), 81 (2018). https://doi.org/10.1007/s00025-018-0843-4
Du, D.D., Zhu, Y.C.: Constructions of K-g-frames and tight K-g-frames in Hilbert spaces. Bull. Malays. Math. Sci. Soc. (2020). https://doi.org/10.1007/s40840-020-00911-0
Găvruţa, L.: Frames for operators. Appl. Comp. Harm. Anal. 32, 139–144 (2012)
Guo, X.X.: Canonical dual \(K\)-Bessel sequences and dual \(K\)-Bessel generators for unitary systems of Hilbert spaces. J. Math. Anal. Appl. 444, 598–609 (2016)
Huang, Y.D., Shi, S.N.: New constructions of \(K\)-g-frames. Results Math. 73, 162 (2018). https://doi.org/10.1007/s00025-018-0924-4
Khosravi, A., Banyarani, J.S.: Weaving g-frames and weaving fusion frames. Bull. Malays. Math. Sci. Soc. (2018). https://doi.org/10.1007/s40840-018-0647-4
Obeidat, S., Samarah, S., Casazza, P.G., Tremain, J.C.: Sums of Hilbert space frames. J. Math. Anal. Appl. 351, 579–585 (2009)
Xiang, Z.Q., Li, Y.M.: Frame sequences and dual frames for operators. ScienceAisa 42, 222–230 (2016)
Xiao, X.C., Zhu, Y.C., Găvruţa, L.: Some properties of \(K\)-frames in Hilbert spaces. Results Math. 63(3), 1243–1255 (2013)
Xiao, X.C., Zhu, Y.C., Shu, Z.B., Ding, M.L.: G-frames with bounded linear operators. Rocky Mt. J. Math. 45(2), 675–693 (2015)
Xiao, X.C., Zhu, Y.C.: Exact \(K\)-g-frames in Hilbert spaces. Results Math. 72(3), 1329–1339 (2017)
Vashisht, L.K., Deepshikha: Weaving properties of generalized continuous frames generated by an iterated function system. J. Geom. Phys. 110, 282–295 (2016)
Acknowledgements
We thank the anonymous referees for valuable suggestions and comments, which lead to a significant improvement of our manuscript. This work is partly supported by the National Natural Science Foundation of China (Grant No. 11901099), the Natural Science Foundation of Fujian Province, China (Grant Nos. 2020J01267 and 2020J01496), and the projects of Xiamen University of Technology (Grant Nos. 40199071 and 50419004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Xiao, X., Yan, K., Zhao, G. et al. Tight K-frames and weaving of K-frames. J. Pseudo-Differ. Oper. Appl. 12, 1 (2021). https://doi.org/10.1007/s11868-020-00371-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11868-020-00371-x