Function theory on the annulus in the dp-norm
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2025
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| _version_ | 1866909811658981376 |
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| author | Agler, Jim Lykova, Zinaida Young, N. J. |
| author_facet | Agler, Jim Lykova, Zinaida Young, N. J. |
| contents | In this paper we shall use realization theory to prove new results about a class of holomorphic functions on an annulus \[R_δ\stackrel{\rm def}{=} \{z \in \mathbb{C}: δ<|z|<1\},\] where $0<δ<1$. The class of functions in question arises in the early work of R. G. Douglas and V. I. Paulsen on the rational dilation of a Hilbert space operator $T$ to a normal operator with spectrum in $\partial R_δ$. Their work suggested the following norm $\|\cdot\|_{\mathrm{dp}}$ on the space $\mathrm{Hol}(R_δ)$ of holomorphic functions on $R_δ$, \[ \|ϕ\|_{\mathrm{dp}} \stackrel{\rm def}{=} \sup\{ \|ϕ(T)\|: \|T\|\leq 1, \|T^{-1} \|\leq 1/δ\ \text{and} \ σ(T)\subseteq R_δ\}.\] By analogy with the classical Schur class of holomorphic functions $\mathcal{S} $ with supremum norm at most $1$ on the disc $\mathbb{D}$, it is natural to consider the dp-Schur class $\mathcal{S}_\mathrm{dp}$ of holomorphic functions of dp-norm at most $1$ on $R_δ$.
Our central result is a Pick interpolation theorem for functions in $\mathcal{S}_\mathrm{dp}$ that is analogous to Abrahamse's Interpolation Theorem for bounded holomorphic functions on a multiply-connected domain. For a tuple $λ=(λ_1,\dots,λ_n)$ of distinct interpolation nodes in $R_δ$, we introduce a special set $\mathcal{G}_{\mathrm {dp}}(λ)$ of positive definite $n\times n$ matrices, which we call DP Szegő kernels. The DP Pick problem $λ_j \mapsto z_j, j=1,\dots,n$, is shown to be solvable if and only if, \[ [(1-\bar z_i z_j)g_{ij}] \ge 0 \; \text{ for all}\; g \in \mathcal{G}_{\mathrm {dp}} (λ).\] We prove further that a solvable DP Pick problem has a solution which is a rational function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04483 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Function theory on the annulus in the dp-norm Agler, Jim Lykova, Zinaida Young, N. J. Complex Variables 47B99, 30E05, 32A26 In this paper we shall use realization theory to prove new results about a class of holomorphic functions on an annulus \[R_δ\stackrel{\rm def}{=} \{z \in \mathbb{C}: δ<|z|<1\},\] where $0<δ<1$. The class of functions in question arises in the early work of R. G. Douglas and V. I. Paulsen on the rational dilation of a Hilbert space operator $T$ to a normal operator with spectrum in $\partial R_δ$. Their work suggested the following norm $\|\cdot\|_{\mathrm{dp}}$ on the space $\mathrm{Hol}(R_δ)$ of holomorphic functions on $R_δ$, \[ \|ϕ\|_{\mathrm{dp}} \stackrel{\rm def}{=} \sup\{ \|ϕ(T)\|: \|T\|\leq 1, \|T^{-1} \|\leq 1/δ\ \text{and} \ σ(T)\subseteq R_δ\}.\] By analogy with the classical Schur class of holomorphic functions $\mathcal{S} $ with supremum norm at most $1$ on the disc $\mathbb{D}$, it is natural to consider the dp-Schur class $\mathcal{S}_\mathrm{dp}$ of holomorphic functions of dp-norm at most $1$ on $R_δ$. Our central result is a Pick interpolation theorem for functions in $\mathcal{S}_\mathrm{dp}$ that is analogous to Abrahamse's Interpolation Theorem for bounded holomorphic functions on a multiply-connected domain. For a tuple $λ=(λ_1,\dots,λ_n)$ of distinct interpolation nodes in $R_δ$, we introduce a special set $\mathcal{G}_{\mathrm {dp}}(λ)$ of positive definite $n\times n$ matrices, which we call DP Szegő kernels. The DP Pick problem $λ_j \mapsto z_j, j=1,\dots,n$, is shown to be solvable if and only if, \[ [(1-\bar z_i z_j)g_{ij}] \ge 0 \; \text{ for all}\; g \in \mathcal{G}_{\mathrm {dp}} (λ).\] We prove further that a solvable DP Pick problem has a solution which is a rational function. |
| title | Function theory on the annulus in the dp-norm |
| topic | Complex Variables 47B99, 30E05, 32A26 |
| url | https://arxiv.org/abs/2505.04483 |