2025/05/07 by Jim Agler, Agler, Jim, Zinaida A. Lykova +3
Mathematics · #Holomorphic and Operator Theory #Analytic and geometric function theory #Advanced Banach Space Theory
paper · pdf · doi:10.48550/arxiv.2505.04483
In this paper we shall use realization theory to prove new results about a class of holomorphic functions on an annulus Rδ\stackrel\rm def= \z ∈ ℂ: δlt;|z|lt;1\, where 0<δ<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 ∂ Rδ. Their work suggested the following norm ‖⋅‖dp on the space Hol(Rδ) of holomorphic functions on Rδ, ‖ϕ‖dp \stackrel\rm def= sup\ ‖ϕ(T)‖: ‖T‖≤ 1, ‖T-1 ‖≤ 1/δ and σ(T)⊆ Rδ\. By analogy with the classical Schur class of holomorphic functions S with supremum norm at most 1 on the disc \mathbbD, it is natural to consider the dp-Schur class Sdp of holomorphic functions of dp-norm at most 1 on Rδ. Our central result is a Pick interpolation theorem for functions in Sdp that is analogous to Abrahamse's Interpolation Theorem for bounded holomorphic functions on a multiply-connected domain. For a tuple λ=(λ1,…,λn) of distinct interpolation nodes in Rδ, we introduce a special set G_\mathrm dp(λ) of positive definite n× n matrices, which we call DP Szegő kernels. The DP Pick problem λj ↦ zj, j=1,…,n, is shown to be solvable if and only if, [(1- zi zj)gij] ≥ 0 for all g ∈ G_\mathrm dp (λ). We prove further that a solvable DP Pick problem has a solution which is a rational function.