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Function theory on the annulus in the dp-norm.

Jim Agler1, Zinaida A Lykova2, N J Young2

  • 1Department of Mathematics, University of California at San Diego, San Diego, 92103 USA.

Integral Equations and Operator Theory
|November 5, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces a new class of holomorphic functions on an annulus, the dp-Schur class, and establishes a Pick interpolation theorem for it. The findings enable the development of extremal function theories analogous to finite Blaschke products.

Keywords:
Crouzeix conjectureDouglas-Paulsen familyHilbert space modelHolomorphic functionsannulus

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Area of Science:

  • Complex Analysis
  • Operator Theory
  • Functional Analysis

Background:

  • Investigates holomorphic functions on an annulus R_δ, extending concepts from the classical Schur class.
  • Builds upon the work of Douglas and Paulsen on rational dilation of Hilbert space operators.
  • Introduces a novel dp-norm for holomorphic functions on R_δ.

Purpose of the Study:

  • To prove new results for a class of holomorphic functions on an annulus R_δ using realization theory.
  • To establish a Pick interpolation theorem for the dp-Schur class of functions.
  • To explore the properties and potential applications of this new function class.

Main Methods:

  • Utilizes realization theory, a technique favored by Rien Kaashoek.
  • Defines and analyzes the dp-norm on the space of holomorphic functions Hol(R_δ).
  • Introduces DP Szegő kernels (G_dp(λ)) for a tuple of interpolation nodes λ.

Main Results:

  • A Pick interpolation theorem is established for the dp-Schur class on R_δ.
  • The solvability condition for the DP Pick problem is determined using DP Szegő kernels.
  • Solvable DP Pick problems yield rational function solutions with finite-dimensional models.

Conclusions:

  • The study introduces and characterizes the dp-Schur class and its interpolation properties.
  • The results pave the way for a theory of extremal functions analogous to finite Blaschke products.
  • This work extends classical function theory to a broader class of functions and domains.