Function theory on the annulus in the dp-norm

Abstract

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 $0<\hat{I}´<1$. 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 $\partial R_{\hat{I}}´$. Their work suggested the following norm $\Vert \cdot {\Vert }_{\mathrm{dp}}$ on the space $\mathrm{Hol}(R_{\hat{I}}´)$ of holomorphic functions on $R_{\hat{I}}´$, \[ \|ϕ\|_{\mathrm{dp}} \stackrel{\rm def}{=} \sup\{ \|ϕ(T)\|: \|T\|\leq 1, \|T^{-1} \|\leq 1/δ\ \text{and} \ σ(T)\subseteq R_δ\}.\] 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 $R_{\hat{I}}´$. 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 $\hat{I}»=(\hat{I}{»}_1,\dots ,\hat{I}{»}_n)$ of distinct interpolation nodes in $R_{\hat{I}}´$, we introduce a special set ${\mathcal{G}}_{\mathrm{dp}}(\hat{I}»)$ of positive definite $n\times n$ matrices, which we call DP Szegő kernels. The DP Pick problem $\hat{I}{»}_j\mapsto z_j,j=1,\dots ,n$, 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.

29 pages. This version is a slight modification of the original paper following a referee report. It will appear in the Journal "Integral Equations and Operator Theory"