Function theory on the annulus in the dp-norm

In this paper we shall use realization theory to prove new results about a class of holomorphic functions on an annulus $$R_{\delta }\{z\in \mathbb{C}:\delta <\mathrm{\mid }z\mathrm{\mid }<1\},$$ 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 $\mathrm{\partial }{R}_{\delta }$. Their work suggested the following norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{d}\mathrm{p}}$ on the space $\mathrm{H}\mathrm{o}\mathrm{l}(R_{\delta })$ of holomorphic functions on $R_{\delta }$, $$\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{d}\mathrm{p}}\sup \{\mathrm{\parallel }\phi (T)\mathrm{\parallel }:\mathrm{\parallel }T\mathrm{\parallel }\le 1,\mathrm{\parallel }{T}^{-1}\mathrm{\parallel }\le 1\mathrm{/}\delta \text{ and }\sigma (T)\subseteq R_{\delta }\}\mathrm{.}$$ 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 ${\mathcal{S}}_{\mathrm{d}\mathrm{p}}$ of holomorphic functions of dp-norm at most $1$ on $R_{\delta }$. Our central result is a Pick interpolation theorem for functions in ${\mathcal{S}}_{\mathrm{d}\mathrm{p}}$ 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 ${\mathcal{G}}_{\mathrm{d}\mathrm{p}}(\lambda )$ of positive definite $n\times n$ matrices, which we call DP Szegő kernels. The DP Pick problem ${\lambda }_j\mapsto z_j,j=1,\dots ,n$, is shown to be solvable if and only if, $$[(1-{\bar{z}}_i{z}_j)g_{ij}]\ge 0\text{ for all}g\in {\mathcal{G}}_{\mathrm{d}\mathrm{p}}(\lambda )\mathrm{.}$$ We prove further that a solvable DP Pick problem has a solution which is a rational function.