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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.17358 |
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| _version_ | 1866912844819202048 |
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| author | Lin, Dexie Wang, Yi |
| author_facet | Lin, Dexie Wang, Yi |
| contents | Given a commuting $n$-tuple of bounded linear operators on a Hilbert space, together with a distinguished cyclic vector, Jim Agler defined a linear functional $Λ_{\mathbf{T},h}$ on the polynomial ring $\mathbb{C}[\mathbf{z},\bar{\mathbf{z}}]$. ``Near subnormality properties'' of an operator $T$ are translated into positivity properties of $Λ_{T,h}$. In this paper, we approach ``near subnormality properties'' in a different way by answering the following question: when is $Λ_{\mathbf{T},h}$ given by a compactly supported distribution? The answer is in terms of the off-diagonal growth condition of a two-variable kernel function $F_{\mathbf{T},h}$ on $\mathbb{C}^n$. Using the reproducing kernel Hilbert spaces (RKHS) defined by the kernel function $F_{\mathbf{T},h}$, we give a function model for all cyclic commuting $n$-tuples. This potentially gives a different approach to operator models. The reproducing kernels of the Fock space are used in the construction of $F_{\mathbf{T},h}$, but one may also replace the Fock space by other RKHS. We give many examples in the last section. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17358 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cyclic Operators, Linear Functionals and RKHS Lin, Dexie Wang, Yi Functional Analysis 47B02 Given a commuting $n$-tuple of bounded linear operators on a Hilbert space, together with a distinguished cyclic vector, Jim Agler defined a linear functional $Λ_{\mathbf{T},h}$ on the polynomial ring $\mathbb{C}[\mathbf{z},\bar{\mathbf{z}}]$. ``Near subnormality properties'' of an operator $T$ are translated into positivity properties of $Λ_{T,h}$. In this paper, we approach ``near subnormality properties'' in a different way by answering the following question: when is $Λ_{\mathbf{T},h}$ given by a compactly supported distribution? The answer is in terms of the off-diagonal growth condition of a two-variable kernel function $F_{\mathbf{T},h}$ on $\mathbb{C}^n$. Using the reproducing kernel Hilbert spaces (RKHS) defined by the kernel function $F_{\mathbf{T},h}$, we give a function model for all cyclic commuting $n$-tuples. This potentially gives a different approach to operator models. The reproducing kernels of the Fock space are used in the construction of $F_{\mathbf{T},h}$, but one may also replace the Fock space by other RKHS. We give many examples in the last section. |
| title | Cyclic Operators, Linear Functionals and RKHS |
| topic | Functional Analysis 47B02 |
| url | https://arxiv.org/abs/2507.17358 |