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_\delta \stackrel{\rm def}{=} {z \in \mathbb{C}: \delta <|z|<1},] where $0<\delta<1$. 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 $\partial R_\delta$. Their work suggested the following norm $|\cdot|{\mathrm{dp}}$ on the space $\mathrm{Hol}(R\delta)$ of holomorphic functions on $R_\delta$, [ |\phi|{\mathrm{dp}} \stackrel{\rm def}{=} \sup{ |\phi(T)|: |T|\leq 1, |T^{-1} |\leq 1/\delta \ \text{and} \ \sigma(T)\subseteq R\delta}.] 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 $R\delta$. Our central result is a generalization of the classical realization formula, for $\phi \in \mathcal{S} $, to functions from $\mathcal{S}\mathrm{dp}$. A second 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 $\lambda=(\lambda_1,\dots,\lambda_n)$ of distinct interpolation nodes in $R_\delta$, we introduce a special set $\mathcal{G}{\mathrm {dp}}(\lambda)$ of positive definite $n\times n$ matrices, which we call DP Szeg\H{o} kernels. The DP Pick problem $\lambda_j \mapsto z_j, j=1,\dots,n$, 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}} (\lambda).] We prove further that a solvable DP Pick problem has a solution which is a rational function.