Contractivity of Möbius functions of operators
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916405671100416 |
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| author | Ransford, Thomas Tsedenbayar, Dashdondog |
| author_facet | Ransford, Thomas Tsedenbayar, Dashdondog |
| contents | Let $T$ be a injective bounded linear operator on a complex Hilbert space. We characterize the complex numbers $λ,μ$ for which $(I+λT)(I+μT)^{-1}$ is a contraction, the characterization being expressed in terms of the numerical range of the possibly unbounded operator $T^{-1}$. When $T=V$, the Volterra operator on $L^2[0,1]$, this leads to a result of Khadkhuu, Zemánek and the second author, characterizing those $λ,μ$ for which $(I+λV)(I+μV)^{-1}$ is a contraction. Taking $T=V^n$, we further deduce that $(I+λV^n)(I+μV^n)^{-1}$ is never a contraction if $n\ge2$ and $λ\neμ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14125 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Contractivity of Möbius functions of operators Ransford, Thomas Tsedenbayar, Dashdondog Functional Analysis Primary 47G10, Secondary 47A12 Let $T$ be a injective bounded linear operator on a complex Hilbert space. We characterize the complex numbers $λ,μ$ for which $(I+λT)(I+μT)^{-1}$ is a contraction, the characterization being expressed in terms of the numerical range of the possibly unbounded operator $T^{-1}$. When $T=V$, the Volterra operator on $L^2[0,1]$, this leads to a result of Khadkhuu, Zemánek and the second author, characterizing those $λ,μ$ for which $(I+λV)(I+μV)^{-1}$ is a contraction. Taking $T=V^n$, we further deduce that $(I+λV^n)(I+μV^n)^{-1}$ is never a contraction if $n\ge2$ and $λ\neμ$. |
| title | Contractivity of Möbius functions of operators |
| topic | Functional Analysis Primary 47G10, Secondary 47A12 |
| url | https://arxiv.org/abs/2409.14125 |