Abstract
In this note, we find a class of unitary operators, denoted by \(\mathcal {U}\), on a complex separable infinite-dimensional Hilbert space \(\mathcal {H}\) such that for any \(U\in \mathcal {U}\), there exists an operator R of rank 1 on \(\mathcal {H}\) such that \(U+R\) is hypercyclic and the hypercyclic vectors are of full measure. Then, these results are applied to the controllability of discrete-time linear control systems, where the rank one perturbation is used as a one-dimensional feedback control law.
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Acknowledgements
The authors wish to thank the anonymous referees for their careful reading and for valuable comments which improved the quality of the manuscript. The first author was supported by the National Natural Science Foundation of China (No. 12001113) and the Educational Commission of Guangdong Province (No. 2019KQNCX096). The second author was supported by National Natural Science Foundation of China (No. 11701584 and 11801096), the Natural Science Research Project of Guangdong Province (No. 2018KTSCX122 and 2017KQNCX122). The authors would like to thank Prof. Yu Huang and Prof. Haiwei Sun for very useful suggestions.
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Liu, X., Chen, Z. Rank One Perturbation of Unitary Operators with Full Measure of Hypercyclic Vectors. Bull. Malays. Math. Sci. Soc. 45, 2475–2492 (2022). https://doi.org/10.1007/s40840-022-01334-9
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DOI: https://doi.org/10.1007/s40840-022-01334-9