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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.05385 |
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| _version_ | 1866912111061368832 |
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| author | Groenewald, Gilbert J. ter Horst, Sanne Woerdeman, Hugo J. |
| author_facet | Groenewald, Gilbert J. ter Horst, Sanne Woerdeman, Hugo J. |
| contents | We show that for a multivariable polynomial $p(z)=p(z_1, \ldots , z_d)$ with a determinantal representation $$ p(z) = p(0) \det (I_n- K (\oplus_{j=1}^d z_j I_{n_j}))$$ the matrix $K$ is structurally similar to a strictly $J$-contractive matrix for some diagonal signature matrix $J$ if and only if the extension of $p(z)$ to a polynomial in $d$-tuples of matrices of arbitrary size given by \[ p(U_1, \ldots , U_d) = p(0,\ldots,0) \det (I_n\otimes I_m- (K\otimes I_m) (\oplus_{j=1}^d I_{n_j}\otimes U_j)), \] where $U_1,\ldots , U_d \in {\mathbb C}^{m \times m}$, $m\in {\mathbb N}$, does not have roots on the noncommutative $d$-torus consisting of $d$-tuples $(U_1, \ldots , U_d)$ of unitary matrices of arbitrary size. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_05385 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Indefinite determinantal representations versus nonsingularities on the noncommutative d-torus Groenewald, Gilbert J. ter Horst, Sanne Woerdeman, Hugo J. Functional Analysis 15A15, 47A13, 13P15 We show that for a multivariable polynomial $p(z)=p(z_1, \ldots , z_d)$ with a determinantal representation $$ p(z) = p(0) \det (I_n- K (\oplus_{j=1}^d z_j I_{n_j}))$$ the matrix $K$ is structurally similar to a strictly $J$-contractive matrix for some diagonal signature matrix $J$ if and only if the extension of $p(z)$ to a polynomial in $d$-tuples of matrices of arbitrary size given by \[ p(U_1, \ldots , U_d) = p(0,\ldots,0) \det (I_n\otimes I_m- (K\otimes I_m) (\oplus_{j=1}^d I_{n_j}\otimes U_j)), \] where $U_1,\ldots , U_d \in {\mathbb C}^{m \times m}$, $m\in {\mathbb N}$, does not have roots on the noncommutative $d$-torus consisting of $d$-tuples $(U_1, \ldots , U_d)$ of unitary matrices of arbitrary size. |
| title | Indefinite determinantal representations versus nonsingularities on the noncommutative d-torus |
| topic | Functional Analysis 15A15, 47A13, 13P15 |
| url | https://arxiv.org/abs/2411.05385 |