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Asymptotic Expansive and Hyperbolic Linear Operators

  • Priyabrata Bag
  • Pabitra Narayan Mandal
  • Pramod Kumar Das
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Abstract

We study the topological dynamical notion of asymptotic expansivity for linear operators. We also introduce two new notions, namely strong asymptotic expansivity and super asymptotic expansivity. We show that all three notions are equivalent to hyperbolicity for linear operators on finite dimensional spaces. In case of infinite-dimension, we construct examples to show that asymptotic expansivity is weaker than strong asymptotic expansivity which is further weaker than super asymptotic expansivity. More interestingly, we prove that super asymptotic expansivity is equivalent to the notion of hyperbolicity. Then, we prove that the super asymptotically expansive linear operators is a dense class in strong operator topology. Finally, we show elegant ways to construct variants of asymptotically expansive linear operators.

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Cite

BibTeXRIS
BibTeX
@misc{bag2026asymptotic,
  title = {{Asymptotic Expansive and Hyperbolic Linear Operators}},
  author = {Priyabrata Bag and Pabitra Narayan Mandal and Pramod Kumar Das},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01196},
  eprint = {2610.01196},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01196},
}
RIS
TY  - UNPB
TI  - Asymptotic Expansive and Hyperbolic Linear Operators
AU  - Priyabrata Bag
AU  - Pabitra Narayan Mandal
AU  - Pramod Kumar Das
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.01196
UR  - https://arxiv.org/abs/2610.01196
AB  - We study the topological dynamical notion of asymptotic expansivity for linear operators. We also introduce two new notions, namely strong asymptotic expansivity and super asymptotic expansivity. We show that all three notions are equivalent to hyperbolicity for linear operators on finite dimensional spaces. In case of infinite-dimension, we construct examples to show that asymptotic expansivity is weaker than strong asymptotic expansivity which is further weaker than super asymptotic expansivity. More interestingly, we prove that super asymptotic expansivity is equivalent to the notion of hyperbolicity. Then, we prove that the super asymptotically expansive linear operators is a dense class in strong operator topology. Finally, we show elegant ways to construct variants of asymptotically expansive linear operators.
ER  -