A sharp bound for the functional calculus of $ρ$-contractions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917855536087040 |
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| author | Schwenninger, Felix L. de Vries, Jens |
| author_facet | Schwenninger, Felix L. de Vries, Jens |
| contents | Let $A$ be a $ρ$-contraction and $f$ a rational function mapping the closed unit disk into itself. With a new characterization of $ρ$-contractions we prove that \begin{align*} \big\|f(A)\big\|\leq \fracρ{2}\big(1-|f(0)|^{2}\big)+\sqrt{\frac{ρ^{2}}{4}\big(1-|f(0)|^{2}\big){}^{2}+|f(0)|^{2}}. \end{align*} We further show that this bound is sharp. This refines an estimate by Okubo--Ando and, for $ρ=2$, is consistent with a result by Drury. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02326 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A sharp bound for the functional calculus of $ρ$-contractions Schwenninger, Felix L. de Vries, Jens Functional Analysis Numerical Analysis Operator Algebras Primary: 47A60. Secondary: 15A60, 47A20 Let $A$ be a $ρ$-contraction and $f$ a rational function mapping the closed unit disk into itself. With a new characterization of $ρ$-contractions we prove that \begin{align*} \big\|f(A)\big\|\leq \fracρ{2}\big(1-|f(0)|^{2}\big)+\sqrt{\frac{ρ^{2}}{4}\big(1-|f(0)|^{2}\big){}^{2}+|f(0)|^{2}}. \end{align*} We further show that this bound is sharp. This refines an estimate by Okubo--Ando and, for $ρ=2$, is consistent with a result by Drury. |
| title | A sharp bound for the functional calculus of $ρ$-contractions |
| topic | Functional Analysis Numerical Analysis Operator Algebras Primary: 47A60. Secondary: 15A60, 47A20 |
| url | https://arxiv.org/abs/2412.02326 |