Some properties of weaving $ K $-frames in $ n $-Hilbert space

Gang Wang; Gang Wang

doi:10.3934/math.20241242

AIMS Mathematics

2024, Volume 9, Issue 9: 25438-25456. doi: 10.3934/math.20241242

Previous Article Next Article

Research article

Some properties of weaving $ K $-frames in $ n $-Hilbert space

Gang Wang ^,

School of mathematical science, Xinjiang Normal University, Urumqi, Xinjiang, 830017, China

Received: 24 July 2024 Revised: 19 August 2024 Accepted: 26 August 2024 Published: 30 August 2024
MSC : 42C15, 42C40, 47D63

$ K $-frames are more generalized than ordinary frames, particularly in terms of their weaving properties. The study of weaving $ K $-frames in Hilbert space has already been explored. Given the significance of $ n $-Hilbert spaces in functional analysis, it is essential to study weaving $ K $-frames in $ n $-Hilbert spaces. In this paper, we introduced the notion of weaving $ K $-frames in $ n $-Hilbert spaces and obtained some new properties for these frames using operator theory methods. First, the concept of weaving $ K $-frames in $ n $-Hilbert spaces is developed, and examples are given. By virtue of auxiliary operators, such as the preframe operator, analysis operator, and frame operator, some new properties and characterizations of these frames are presented, and several new methods for their construction are given. Stability and perturbation results are discussed and new inequalities are established as applications.
- $ n $-Hilbert space,
- weaving frames,
- weaving $ K $-frames,
- $ K $-woven frames,
- perturbation
Citation: Gang Wang. Some properties of weaving $ K $-frames in $ n $-Hilbert space[J]. AIMS Mathematics, 2024, 9(9): 25438-25456. doi: 10.3934/math.20241242

Related Papers:

Abstract

$ K $-frames are more generalized than ordinary frames, particularly in terms of their weaving properties. The study of weaving $ K $-frames in Hilbert space has already been explored. Given the significance of $ n $-Hilbert spaces in functional analysis, it is essential to study weaving $ K $-frames in $ n $-Hilbert spaces. In this paper, we introduced the notion of weaving $ K $-frames in $ n $-Hilbert spaces and obtained some new properties for these frames using operator theory methods. First, the concept of weaving $ K $-frames in $ n $-Hilbert spaces is developed, and examples are given. By virtue of auxiliary operators, such as the preframe operator, analysis operator, and frame operator, some new properties and characterizations of these frames are presented, and several new methods for their construction are given. Stability and perturbation results are discussed and new inequalities are established as applications.

References

[1]	X. Xiao, Y. Zhu, L. Găvruţa, Some properties of K-frames in Hilbert spaces, Results Math., 63 (2013), 1243–1255. https://doi.org/10.1007/s00025-012-0266-6 doi: 10.1007/s00025-012-0266-6
[2]	L. Găvruţa, Frames for operators, Appl. Comput. Harmon. Anal., 32 (2012), 139–144. https://doi.org/10.1016/j.acha.2011.07.006
[3]	T. Bemrose, P. G. Cassaza, K. Grochenig, M. C. Lammers, R. G. Lynch, Weaving frames, Oper. Matrices, 10 (2016), 1093–1116. https://doi.org/10.7153/oam-10-61
[4]	Deepshikha, L. K. Vashisht, Weaving $K$-frames in Hilbert spaces, Results Math., 73 (2018), 1–20. https://doi.org/10.1007/s00025-018-0843-4 doi: 10.1007/s00025-018-0843-4
[5]	Z. Xiang, Some new results of Weaving K-frames in Hilbert space, Numer. Func. Anal. Opt., 42 (2021), 409–429. https://doi.org/10.1080/01630563.2021.1882488. doi: 10.1080/01630563.2021.1882488
[6]	S. Gähler, Lineare 2-Normierte Raume, Math. Nachr., 28 (1965), 1–43. https://doi.org/10.1002/mana.19640280102
[7]	C. Diminnie, S. Gähler, A. White, 2-Inner product spaces, Demonstratio Math., 10 (1977), 169–188.
[8]	H. Gunawan, M. Mashadi, On n-normed spaces, Int. J. Math. Math. Sci., 27 (2001), 631–639. https://doi.org/10.1155/S0161171201010675
[9]	A. Misiak, n‐Inner product spaces, Math. Nachr., 140 (1989), 299–319.
[10]	A. Arefijamaal, S. Ghadir, Frames in 2-Inner product spaces, Iran. J. Math. Sci. Inform., 8 (2013), 123–130.
[11]	P. Ghosh, T. Samanta, Construction of frame relative to n-Hilbert space, J. Linear Topol. Algebra, 10 (2021), 117–130. https://doi.org/10.48550/arXiv.2101.01657 doi: 10.48550/arXiv.2101.01657
[12]	P. Ghosh, T. Samanta, Some properties of K-frame in n-Hilbert space, arXiv preprint, 2021. https://doi.org/10.48550/arXiv.2102.05255
[13]	D. Li, J. S. Leng, T. Huang, X. Li, On weaving g-frames for Hilbert spaces, Complex Anal. Oper. Th., 14 (2020), 1–25. https://doi.org/10.1007/s11785-020-00991-7 doi: 10.1007/s11785-020-00991-7
[14]	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/10.1090/S0002-9939-1966-0203464-1 doi: 10.1090/S0002-9939-1966-0203464-1
[15]	O. Christensen, An introduction to frames and Riesz bases, Appl. Numer. Harmon. Anal., Birkhauser: Cham, Switzerland, 2016. https://doi.org/10.1007/978-3-319-25613-9
[16]	H. Gunawan, On convergence in n-inner product spaces, Bull. Malays. Math. Sci. So., 25 (2002), 11–16.
[17]	A. Misiak, Orthogonality and orthonormality in n‐inner product spaces, Math. Nach., 143 (1989), 249–261. https://doi.org/10.1002/mana.19891430119 doi: 10.1002/mana.19891430119
[18]	H. Gunawan, The space of p-summable sequences and its natural n-norm, Bull. Aust. Math. Soc., 64 (2001), 137–147.
[19]	H. Gunawan, An inner product that makes a set of vectors orthonormal, Austral. Math. Soc. Gaz., 28 (2001), 194–197.
[20]	H. Gunawan, Inner products on n-inner product spaces, Soochow J. Math., 28 (2002), 389–398.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)