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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2401.15591 |
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| _version_ | 1866914656033964032 |
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| author | Bhattacharyya, Tirthankar Jindal, Abhay |
| author_facet | Bhattacharyya, Tirthankar Jindal, Abhay |
| contents | We consider a unitarily invariant complete Nevanlinna-Pick kernel denoted by $s$ and a commuting $d$-tuple of bounded operators $T = (T_{1}, \dots, T_{d})$ satisfying a natural contractivity condition with respect to $s$. We associate with $T$ its curvature invariant which is a non-negative real number bounded above by the dimension of a defect space of $\bfT$. The instrument which makes this possible is the characteristic function developed in \cite{BJ}.
\medskip
We present an asymptotic formula for the curvature invariant. In the special case when $\bfT$ is pure, we provide a notably simpler formula, revealing that in this instance, the curvature invariant is an integer. We further investigate its connection with an algebraic invariant known as fibre dimension. Moreover, we obtain a refined and simplified asymptotic formula for the curvature invariant of $\bfT$ specifically when its characteristic function is a polynomial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15591 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complete Nevanlinna-Pick Kernels and the Curvature Invariant Bhattacharyya, Tirthankar Jindal, Abhay Functional Analysis 47A13, 47A15, 46E22 We consider a unitarily invariant complete Nevanlinna-Pick kernel denoted by $s$ and a commuting $d$-tuple of bounded operators $T = (T_{1}, \dots, T_{d})$ satisfying a natural contractivity condition with respect to $s$. We associate with $T$ its curvature invariant which is a non-negative real number bounded above by the dimension of a defect space of $\bfT$. The instrument which makes this possible is the characteristic function developed in \cite{BJ}. \medskip We present an asymptotic formula for the curvature invariant. In the special case when $\bfT$ is pure, we provide a notably simpler formula, revealing that in this instance, the curvature invariant is an integer. We further investigate its connection with an algebraic invariant known as fibre dimension. Moreover, we obtain a refined and simplified asymptotic formula for the curvature invariant of $\bfT$ specifically when its characteristic function is a polynomial. |
| title | Complete Nevanlinna-Pick Kernels and the Curvature Invariant |
| topic | Functional Analysis 47A13, 47A15, 46E22 |
| url | https://arxiv.org/abs/2401.15591 |