Some results on frames by pre-frame operators in Q-Hilbert spaces

Yan Ling Fu; Wei Zhang; Yan Ling Fu; Wei Zhang

doi:10.3934/math.20231480

AIMS Mathematics

2023, Volume 8, Issue 12: 28878-28896. doi: 10.3934/math.20231480

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Some results on frames by pre-frame operators in Q-Hilbert spaces

Yan Ling Fu ¹,
Wei Zhang ^{2
,
,}

1.
School of Statistics and Mathematics, Henan Finance University, ZhengZhou 450046, China
2.
School of Mathematics and Information Sciences, Henan University of Economics and Law, Zhengzhou 450046, China

Received: 17 August 2023 Revised: 16 October 2023 Accepted: 17 October 2023 Published: 24 October 2023
MSC : 47A05, 42C15

Quaternionic Hilbert (Q-Hilbert) spaces are frequently used in applied physical sciences and especially in quantum physics. In order to solve some problems of many nonlinear physical systems, the frame theory of Q-Hilbert spaces was studied. Frames in Q-Hilbert spaces not only retain the frame properties, but also have some advantages, such as a simple structure for approximation. In this paper, we first characterized Hilbert (orthonormal) bases, frames, dual frames and Riesz bases, and obtained the accurate expressions of all dual frames of a given frame by taking advantage of pre-frame operators. Second, we discussed the constructions of frames with the help of the pre-frame operators and gained some more general methods to construct new frames. Moreover, we obtained a necessary and sufficient condition for the finite sum of frames to be a (tight) frame, and the obtained results further enriched and improved the frame theory of the Q-Hilbert space.
- frame,
- pre-frame operator,
- dual frame,
- sum of frames
Citation: Yan Ling Fu, Wei Zhang. Some results on frames by pre-frame operators in Q-Hilbert spaces[J]. AIMS Mathematics, 2023, 8(12): 28878-28896. doi: 10.3934/math.20231480

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Abstract

Quaternionic Hilbert (Q-Hilbert) spaces are frequently used in applied physical sciences and especially in quantum physics. In order to solve some problems of many nonlinear physical systems, the frame theory of Q-Hilbert spaces was studied. Frames in Q-Hilbert spaces not only retain the frame properties, but also have some advantages, such as a simple structure for approximation. In this paper, we first characterized Hilbert (orthonormal) bases, frames, dual frames and Riesz bases, and obtained the accurate expressions of all dual frames of a given frame by taking advantage of pre-frame operators. Second, we discussed the constructions of frames with the help of the pre-frame operators and gained some more general methods to construct new frames. Moreover, we obtained a necessary and sufficient condition for the finite sum of frames to be a (tight) frame, and the obtained results further enriched and improved the frame theory of the Q-Hilbert space.

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