# Function theory on the annulus in the dp-norm

## TL;DR

This paper develops a new theory of holomorphic functions on an annulus using a special norm and proves a Pick interpolation theorem for these functions.

## Contribution

The paper introduces a Pick interpolation theorem for the dp-Schur class of holomorphic functions on an annulus.

## Key findings

- A Pick interpolation theorem is established for functions in the dp-Schur class on an annulus.
- A solvable DP Pick problem has a rational solution with a finite-dimensional model.
- The DP Szegő kernels are defined and used to characterize solvability of the DP Pick problem.

## Abstract

In this paper we shall use realization theory, a favourite technique of Rien Kaashoek, to prove new results about a class of holomorphic functions on an annulus \documentclass[12pt]{minimal}
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\begin{document}$$ R_\delta {\mathop {=}\limits ^\textrm{def}}\{z\in \mathbb {C}: \delta<|z|<1\}, $$\end{document}Rδ=def{z∈C:δ<|z|<1},where \documentclass[12pt]{minimal}
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\begin{document}$$0<\delta <1$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$\partial R_\delta $$\end{document}∂Rδ. Their work suggested the following norm \documentclass[12pt]{minimal}
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\begin{document}$$\Vert \cdot \Vert _{\textrm{dp}}$$\end{document}‖·‖dp on the space \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{Hol}(R_\delta )$$\end{document}Hol(Rδ) of holomorphic functions on \documentclass[12pt]{minimal}
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\begin{document}$$R_\delta $$\end{document}Rδ, \documentclass[12pt]{minimal}
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\begin{document}$$ \Vert \varphi \Vert _{\textrm{dp}} {\mathop {=}\limits ^\textrm{def}} \sup \{ \Vert \varphi (T)\Vert : \Vert T\Vert \le 1, \Vert T^{-1}\Vert \le 1/\delta \ \text {and} \ \sigma (T)\subseteq R_\delta \}. $$\end{document}‖φ‖dp=defsup{‖φ(T)‖:‖T‖≤1,‖T-1‖≤1/δandσ(T)⊆Rδ}.By analogy with the classical Schur class of holomorphic functions \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {S} $$\end{document}S with supremum norm at most 1 on the disc \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {D}$$\end{document}D, it is natural to consider the dp-Schur class
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\begin{document}$$\mathcal {S}_\textrm{dp}$$\end{document}Sdp of holomorphic functions of dp-norm at most 1 on \documentclass[12pt]{minimal}
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\begin{document}$$R_\delta $$\end{document}Rδ. Our central result is a Pick interpolation theorem for functions in \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {S}_\textrm{dp}$$\end{document}Sdp that is analogous to Abrahamse’s Interpolation Theorem for bounded holomorphic functions on a multiply-connected domain. For a tuple \documentclass[12pt]{minimal}
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\begin{document}$$\lambda =(\lambda _1,\dots ,\lambda _n)$$\end{document}λ=(λ1,⋯,λn) of distinct interpolation nodes in \documentclass[12pt]{minimal}
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\begin{document}$$R_\delta $$\end{document}Rδ, we introduce a special set \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {G}_\textrm{dp}(\lambda )$$\end{document}Gdp(λ) of positive definite \documentclass[12pt]{minimal}
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\begin{document}$$n\times n$$\end{document}n×n matrices, which we call DP Szegő kernels. The DP Pick problem \documentclass[12pt]{minimal}
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\begin{document}$$\lambda _j \mapsto z_j, j=1,\dots ,n$$\end{document}λj↦zj,j=1,⋯,n, is shown to be solvable if and only if, \documentclass[12pt]{minimal}
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\begin{document}$$ {[}(1-{\overline{z}}_i z_j)g_{ij}] \ge 0 \; \text { for all}\; g \in \mathcal {G}_{\textrm{dp}} (\lambda ). $$\end{document}[(1-z¯izj)gij]≥0for allg∈Gdp(λ).We prove further that a solvable DP Pick problem has a solution which is a rational function with a finite-dimensional model, an intriguing result which opens up the possibility of a theory of extremal functions from \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {S}_\textrm{dp}$$\end{document}Sdp analogous to the theory of finite Blaschke products.