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Dynamical classification for complex matrices

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Abstract

In this paper, we study the noncommutative functional equation \(h(\lambda z)-\lambda {h(z)}=g(z),~z\in {\mathbb {C}}\) and we give a new perspective from this equation to obtain a completely dynamical classification for complex matrices. Coarsely speaking, there are four different types: \(0,\frac{1}{2},2\) and \(e^{{\mathbf {i}}2\pi \theta }\) with \(\theta \in [0,\frac{1}{2}]\) for diagonal matrices and Jordan matrices, respectively. Moreover, we obtain that for a complex matrix A, if its eigenvalues are \(0<|\lambda _i|\ne 1\), where \(1\le i\le \mathrm{rank}(A)\), then A is topologically conjugate to a diagonal matrix on \({\mathbb {C}}^{\mathrm{rank}(A)}\).

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Acknowledgements

This paper is supported by the National Nature Science Foundation of China (Grant no. 11801428). Words are powerless to express my gratitude to Editorial Board for their helpful suggestions and kindly patience. I would like to thank the referee for his/her careful reading of the paper and helpful comments and suggestions. Also, I would like to show my deepest gratitude to Prof. Xiaoman Chen, Yijun Yao and Jiawen Zhang for their helpful suggestions in the completion of this paper. Lastly, I shall extend my thanks to all those who have offered their help to me.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Lvlin Luo

  2. Arts and Sciences Teaching Department, Shanghai University of Medicine and Health Sciences, Shanghai, 201318, China

    Lvlin Luo

  3. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, China

    Lvlin Luo

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  1. Lvlin Luo
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Correspondence to Lvlin Luo.

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Communicated by Manuel D. Contreras.

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Luo, L. Dynamical classification for complex matrices. Ann. Funct. Anal. 12, 20 (2021). https://doi.org/10.1007/s43034-020-00106-5

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  • DOI: https://doi.org/10.1007/s43034-020-00106-5

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