On the Crouzeix ratio for $N\times N$ matrices
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913512450686976 |
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| author | Malman, Bartosz Mashreghi, Javad O'Loughlin, Ryan Ransford, Thomas |
| author_facet | Malman, Bartosz Mashreghi, Javad O'Loughlin, Ryan Ransford, Thomas |
| contents | The Crouzeix ratio $ψ(A)$ of an $N\times N$ complex matrix $A$ is the supremum of $\|p(A)\|$ taken over all polynomials $p$ such that $|p|\le 1$ on the numerical range of $A$. It is known that $ψ(A)\le 1+\sqrt{2}$, and it is conjectured that $ψ(A)\le 2$. In this note, we show that $ψ(A)\le C_N$, where $C_N$ is a constant depending only on $N$ and satisfying $C_N<1+\sqrt{2}$. The proof is based on a study of the continuity properties of the map $A\mapsto ψ(A)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_14127 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Crouzeix ratio for $N\times N$ matrices Malman, Bartosz Mashreghi, Javad O'Loughlin, Ryan Ransford, Thomas Functional Analysis Primary 15A60, Secondary 47A12, 47A30 The Crouzeix ratio $ψ(A)$ of an $N\times N$ complex matrix $A$ is the supremum of $\|p(A)\|$ taken over all polynomials $p$ such that $|p|\le 1$ on the numerical range of $A$. It is known that $ψ(A)\le 1+\sqrt{2}$, and it is conjectured that $ψ(A)\le 2$. In this note, we show that $ψ(A)\le C_N$, where $C_N$ is a constant depending only on $N$ and satisfying $C_N<1+\sqrt{2}$. The proof is based on a study of the continuity properties of the map $A\mapsto ψ(A)$. |
| title | On the Crouzeix ratio for $N\times N$ matrices |
| topic | Functional Analysis Primary 15A60, Secondary 47A12, 47A30 |
| url | https://arxiv.org/abs/2409.14127 |