Admissibility and Frame Homotopy for Quaternionic Frames
Admissibility and Frame Homotopy for Quaternionic Frames
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DOI：
10.1016/j.laa.2022.03.023
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发表时间：
2021-08
影响因子：
1.1
通讯作者：
Tom Needham;C. Shonkwiler
中科院分区：
文献类型：
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作者：
Tom Needham;C. Shonkwiler
考虑了以下问题：当给定框架谱和框架向量范数时，什么时候存在四元数框架？当这样的框架存在时，是否总是可以在任何两个框架之间进行插值，同时固定它们的谱和范数？换句话说，第一个问题是四元数框架的容许性问题，第二个问题是框架同伦猜想的推广。我们对这两个问题都给出了完整的答案。对于第一个问题，存在性准则与真实的和复杂的情况完全相同。对于第二种情形，具有特定框架谱和框架向量范数的四元数框架的非空空间总是路径连通的，就像复情形一样。我们证明这些结果的策略是基于解释等价类的框架与给定的框架谱的伴随轨道，这是一种方法，也非常适合于研究真实的和复杂的框架。
We consider the following questions: when do there exist quaternionic frames with given frame spectrum and given frame vector norms? When such frames exist, is it always possible to interpolate between any two while fixing their spectra and norms? In other words, the first question is the admissibility question for quaternionic frames and the second is a generalization of the frame homotopy conjecture. We give complete answers to both questions. For the first question, the existence criterion is exactly the same as in the real and complex cases. For the second, the non-empty spaces of quaternionic frames with specified frame spectrum and frame vector norms are always path-connected, just as in the complex case. Our strategy for proving these results is based on interpreting equivalence classes of frames with given frame spectrum as adjoint orbits, which is an approach that is also well-suited to the study of real and complex frames.