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PreprintCC BY-NC-SA 4.0arXiv

Can one feel the existence of a non-trivial invariant measure?

  • Yannic Steenbeck
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Abstract

It is shown that a bounded linear map on a complex separable Hilbert space with non-trivial invariant probability measures doesn't have to possess eigenvalues. This resolves a question indicated by Flytzanis in 1995 and concretely asked by Grivaux--López-Martínez from 2023 resp. Grivaux--Matheron--Menet from 2021. Still, as a positive result, we prove that every bounded linear operator on a separable complex Banach space for which a non-fixing invariant probability measure exists, has to have some approximate point spectrum on the unit circle minus $1$.

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Cite

BibTeXRIS
BibTeX
@misc{steenbeck2026feel,
  title = {{Can one feel the existence of a non-trivial invariant measure?}},
  author = {Yannic Steenbeck},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05735},
  eprint = {2609.05735},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05735},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Can one feel the existence of a non-trivial invariant measure?
AU  - Yannic Steenbeck
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05735
UR  - https://arxiv.org/abs/2609.05735
AB  - It is shown that a bounded linear map on a complex separable Hilbert space with non-trivial invariant probability measures doesn't have to possess eigenvalues. This resolves a question indicated by Flytzanis in 1995 and concretely asked by Grivaux--López-Martínez from 2023 resp. Grivaux--Matheron--Menet from 2021. Still, as a positive result, we prove that every bounded linear operator on a separable complex Banach space for which a non-fixing invariant probability measure exists, has to have some approximate point spectrum on the unit circle minus $1$.
ER  -