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Almost-invariant half-spaces of operators on ω

  • Noémie Fougnies
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Abstract

We study the Almost-Invariant Subspace Problem for operators defined on non-normable Fréchet spaces and especially on ω. This problem is stated as follows: "Given a bounded operator T on an infinite-dimensional complex Fréchet space X, can we always find a closed almost-invariant half-space?". In this paper we solve the problem for the space ω by showing that an operator T on ω possesses a closed almost-invariant half-space if and only if T is conjugate to F + R + λId or to B + R + λId where F is the Forward shift and B is the Backward shift and where R is a finite rank operator and λ∈ ℂ. We also investigate the case of operators defined on X ⊕ ω where X is an infinite-dimensional Fréchet space with a continuous norm and we show that in this case, every operator defined on X ⊕ ω possesses a closed almost-invariant half-space.

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Cite

BibTeXRIS
BibTeX
@misc{fougnies2026almostinvariant,
  title = {{Almost-invariant half-spaces of operators on $ω$}},
  author = {Noémie Fougnies},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03468},
  eprint = {2609.03468},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03468},
}
RIS
TY  - UNPB
TI  - Almost-invariant half-spaces of operators on ω
AU  - Noémie Fougnies
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03468
UR  - https://arxiv.org/abs/2609.03468
AB  - We study the Almost-Invariant Subspace Problem for operators defined on non-normable Fréchet spaces and especially on ω. This problem is stated as follows: "Given a bounded operator T on an infinite-dimensional complex Fréchet space X, can we always find a closed almost-invariant half-space?". In this paper we solve the problem for the space ω by showing that an operator T on ω possesses a closed almost-invariant half-space if and only if T is conjugate to F + R + λId or to B + R + λId where F is the Forward shift and B is the Backward shift and where R is a finite rank operator and λ∈ ℂ. We also investigate the case of operators defined on X ⊕ ω where X is an infinite-dimensional Fréchet space with a continuous norm and we show that in this case, every operator defined on X ⊕ ω possesses a closed almost-invariant half-space.
ER  -