| 张 伟,李云章.四元数Hilbert空间中Riesz基的刻画*[J].数学年刊A辑,2023,44(1):97~112 |
| 四元数Hilbert空间中Riesz基的刻画* |
| Characterizations of Riesz Bases in Quaternionic Hilbert Spaces |
| Received:October 12, 2020 Revised:November 03, 2022 |
| DOI：10.16205/j.cnki.cama.2023.0008 |
| 中文关键词: 四元数Hilbert空间, 框架, Riesz基, 完备性 |
| 英文关键词:Quaternionic Hilbert spaces, Frames, Riesz bases, Completeness |
| 基金项目:国家自然科学基金(No.11971043), 河南省高等学校重点科研项目(No.21A110004)和河南省科技攻关项目(No.222102210335) |
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| 中文摘要: |
| 四元数Hilbert空间在应用物理科学特别是量子物理中占有重要地位.本文讨论四元数Hilbert空间的框架理论, 在四元数Hilbert空间中引入了Riesz基的概念, 在此基础上刻画了Riesz基,给出了它们的一些等价条件; 特别地, 得到了四元数Hilbert空间中的一个序列是Riesz基的充要条件是它是一个具有双正交序列的完备Bessel序列,且它的双正交序列也是一个完备Bessel序列; 并进一步证明了双正交序列中一个序列的完备性可以从特征刻画中去除.文中举例说明了双正交性、完备性和Bessel性质之间的关系. |
| 英文摘要: |
| Quaternionic Hilbert spaces play an important role in applied physical sciences especially in quantum physics. This paper addresses the frame theory in quaternionic Hilbert spaces. The authors introduce the notion of Riesz bases in quaternionic Hilbert spaces.Then they characterize Riesz bases in this setting, present their some equivalent conditions;particularly, they obtain that a sequence in quaternionic Hilbert spaces is a Riesz basis if and if only it is a complete Bessel sequence with biorthogonal sequence which is also a complete Bessel sequence; and further prove that the completeness of one (any one) of the biorthogonal sequences can be removed from the characterization. Some examples are given to illustrate the relations between biorthogonality, completeness and Bessel property. |
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