Function theory on the annulus in the dp-norm

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 $R_{\delta }\stackrel{\text{def}}{=}\{z\in C:\delta <|z|<1\},$ 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 ${\Vert ·\Vert }_{\text{dp}}$ on the space $\text{Hol}\left(R_{\delta }\right)$ of holomorphic functions on $R_{\delta }$ , ${\Vert \phi \Vert }_{\text{dp}}\stackrel{\text{def}}{=}sup\{\Vert \phi \left(T\right)\Vert :\Vert T\Vert \le 1,\Vert T^{-1}\Vert \le 1/\delta \text{and}\sigma \left(T\right)\subseteq R_{\delta }\}.$ By analogy with the classical Schur class of holomorphic functions $S$ with supremum norm at most 1 on the disc $D$ , it is natural to consider the dp-Schur class $S_{\text{dp}}$ of holomorphic functions of dp-norm at most 1 on $R_{\delta }$ . Our central result is a Pick interpolation theorem for functions in $S_{\text{dp}}$ that is analogous to Abrahamse's Interpolation Theorem for bounded holomorphic functions on a multiply-connected domain. For a tuple $\lambda =({\lambda }_1,\cdots ,{\lambda }_n)$ of distinct interpolation nodes in $R_{\delta }$ , we introduce a special set $G_{\text{dp}}\left(\lambda \right)$ of positive definite $n\times n$ matrices, which we call DP Szegő kernels. The DP Pick problem ${\lambda }_j\mapsto z_j,j=1,\cdots ,n$ , is shown to be solvable if and only if, $[(1-{\overline{z}}_i{z}_j)g_{\mathrm{ij}}]\ge 0\text{for all}g\in G_{\text{dp}}\left(\lambda \right).$ 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 $S_{\text{dp}}$ analogous to the theory of finite Blaschke products.