Crouzeix's conjecture for classes of matrices
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866917556777910272 |
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| author | O'Loughlin, Ryan Virtanen, Jani |
| author_facet | O'Loughlin, Ryan Virtanen, Jani |
| contents | For a matrix $A$ which satisfies Crouzeix's conjecture, we construct several classes of matrices from $A$ for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that Crouzeix's Conjecture holds in full generality if and only if it holds for the differentiation operator on a class of analytic functions. We pose several open questions, which if proved, will prove Crouzeix's conjecture. We also begin an investigation into Crouzeix's conjecture for symmetric matrices and in the case of $3 \times 3$ matrices, we show Crouzeix's conjecture holds for symmetric matrices if and only if it holds for analytic truncated Toeplitz operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_12183 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Crouzeix's conjecture for classes of matrices O'Loughlin, Ryan Virtanen, Jani Functional Analysis Numerical Analysis 5A60, 15A39, 30H10, 47A30 For a matrix $A$ which satisfies Crouzeix's conjecture, we construct several classes of matrices from $A$ for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that Crouzeix's Conjecture holds in full generality if and only if it holds for the differentiation operator on a class of analytic functions. We pose several open questions, which if proved, will prove Crouzeix's conjecture. We also begin an investigation into Crouzeix's conjecture for symmetric matrices and in the case of $3 \times 3$ matrices, we show Crouzeix's conjecture holds for symmetric matrices if and only if it holds for analytic truncated Toeplitz operators. |
| title | Crouzeix's conjecture for classes of matrices |
| topic | Functional Analysis Numerical Analysis 5A60, 15A39, 30H10, 47A30 |
| url | https://arxiv.org/abs/2306.12183 |