A Note on Injective g-Frames in Quaternionic Hilbert Spaces

Zhang, Jianxia; Gao, Fugen; Hong, Guoqing

doi:10.3390/axioms13120851

Open AccessArticle

A Note on Injective g-Frames in Quaternionic Hilbert Spaces

by

Jianxia Zhang

^1,2,

Fugen Gao

² and

Guoqing Hong

^3,*

¹

School of Intelligent Engineering, Henan Institute of Technology, Xinxiang 453003, China

²

School of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China

³

School of Science, Henan Institute of Technology, Xinxiang 453003, China

^*

Author to whom correspondence should be addressed.

Axioms 2024, 13(12), 851; https://doi.org/10.3390/axioms13120851

Submission received: 6 November 2024 / Revised: 1 December 2024 / Accepted: 2 December 2024 / Published: 3 December 2024

(This article belongs to the Section Geometry and Topology)

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Abstract

Motivated by recent advancements in the quantum detection problem employing both discrete and continuous frames, this paper delves into a quantum detection problem utilizing g-frames within the context of quaternionic Hilbert spaces. We offer several equivalent representations of injective g-frames in separable quaternionic Hilbert spaces. By normalizing the trace, we establish a classification for the g-frame injectivity problem. Additionally, we propose a method to derive an injective g-frame by leveraging an injective frame within quaternionic Hilbert spaces. Furthermore, we demonstrate that the injectivity of a g-frame remains intact under a linear isomorphism, while injective g-frames exhibit instability in infinite-dimensional scenarios.

Keywords: g-frame; quantum injectivity; quaternionic Hilbert space; quantum detection

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MDPI and ACS Style

Zhang, J.; Gao, F.; Hong, G. A Note on Injective g-Frames in Quaternionic Hilbert Spaces. Axioms 2024, 13, 851. https://doi.org/10.3390/axioms13120851

AMA Style

Zhang J, Gao F, Hong G. A Note on Injective g-Frames in Quaternionic Hilbert Spaces. Axioms. 2024; 13(12):851. https://doi.org/10.3390/axioms13120851

Chicago/Turabian Style

Zhang, Jianxia, Fugen Gao, and Guoqing Hong. 2024. "A Note on Injective g-Frames in Quaternionic Hilbert Spaces" Axioms 13, no. 12: 851. https://doi.org/10.3390/axioms13120851

APA Style

Zhang, J., Gao, F., & Hong, G. (2024). A Note on Injective g-Frames in Quaternionic Hilbert Spaces. Axioms, 13(12), 851. https://doi.org/10.3390/axioms13120851

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A Note on Injective g-Frames in Quaternionic Hilbert Spaces

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