Typical properties of positive contractions and the invariant subspace problem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912726753738752 |
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| author | Gillet, Valentin |
| author_facet | Gillet, Valentin |
| contents | In this paper, we first study some elementary properties of a typical positive contraction on $\ell_q$ for the Strong Operator Topology and the Strong* Operator Topology. Using these properties, we prove that a typical positive contraction on $\ell_1$ (resp. on $\ell_2$) has a non-trivial invariant subspace for the Strong Operator Topology (resp. for the Strong Operator Topology and the Strong* Operator Topology). We then focus on the case where $X$ is a Banach space with a basis. We prove that a typical positive contraction on a Banach space with an unconditional basis has no non-trivial closed invariant ideals for the Strong Operator Topology and the Strong* Operator Topology. In particular, this shows that when $X = \ell_q$ with $1 \leq q < \infty$, a typical positive contraction $T$ on $X$ for the Strong Operator Topology (resp. for the Strong* Operator Topology when $1 < q < \infty$) does not satisfy the Abramovich, Aliprantis and Burkinshaw criterion, that is, there is no non-zero positive operator in the commutant of $T$ which is quasinilpotent at a non-zero positive vector of $X$. Finally, we prove that, for the Strong* Operator Topology, a typical positive contraction on a reflexive Banach space with a monotone basis does not satisfy the Abramovich, Aliprantis and Burkinshaw criterion. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_14481 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Typical properties of positive contractions and the invariant subspace problem Gillet, Valentin Functional Analysis 46A45, 47A15, 54E52, 47B65 In this paper, we first study some elementary properties of a typical positive contraction on $\ell_q$ for the Strong Operator Topology and the Strong* Operator Topology. Using these properties, we prove that a typical positive contraction on $\ell_1$ (resp. on $\ell_2$) has a non-trivial invariant subspace for the Strong Operator Topology (resp. for the Strong Operator Topology and the Strong* Operator Topology). We then focus on the case where $X$ is a Banach space with a basis. We prove that a typical positive contraction on a Banach space with an unconditional basis has no non-trivial closed invariant ideals for the Strong Operator Topology and the Strong* Operator Topology. In particular, this shows that when $X = \ell_q$ with $1 \leq q < \infty$, a typical positive contraction $T$ on $X$ for the Strong Operator Topology (resp. for the Strong* Operator Topology when $1 < q < \infty$) does not satisfy the Abramovich, Aliprantis and Burkinshaw criterion, that is, there is no non-zero positive operator in the commutant of $T$ which is quasinilpotent at a non-zero positive vector of $X$. Finally, we prove that, for the Strong* Operator Topology, a typical positive contraction on a reflexive Banach space with a monotone basis does not satisfy the Abramovich, Aliprantis and Burkinshaw criterion. |
| title | Typical properties of positive contractions and the invariant subspace problem |
| topic | Functional Analysis 46A45, 47A15, 54E52, 47B65 |
| url | https://arxiv.org/abs/2409.14481 |