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 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.
Categories: math.CV