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Prediction theory in Hilbert Spaces: Operator-valued Szego Theory

Badr Missaoui, Nicholas H. Bingham

TL;DR

This work extends Szegő theory to operator-valued orthogonal polynomials on the unit circle in infinite-dimensional settings, enriching prediction theory for wide-sense stationary and nonstationary processes. It introduces an operator-valued OPUC framework via moments, Schur complements, and a Schur-structured Verblunsky recursion, then develops operator versions of Christoffel-Darboux kernels and Bernstein-Szegő approximations. The main result is an operator Szegő limit theorem relating Toeplitz-determinant regularizations to the product over Verblunsky coefficients, with a Wiener-algebra regularity condition ensuring the logarithmic determinant integral is well-defined. The findings generalize matrix-valued theory and provide a robust toolkit for analyzing spectral and predictive properties of large-scale or continuous-time operator-valued processes. ${$\displaystyle \lim_{n\to\infty} \mathrm{det}_{2}(T^{R}_{n})/\mathrm{det}_{2}(T^{R}_{n-1}) = \mathrm{det}\prod_{k=0}^{\infty} (I - \alpha_k \alpha_k^{\dagger}) = \exp\left( \frac{1}{2\pi} \int_{-\pi}^{\pi} \log \det \mu'(\theta) \, d\theta \right)$.}$

Abstract

In this paper, we extend some classical results of the Szego theory of orthogonal polynomials on the unit circle to the infinite-dimensional case, and we establish the corresponding Szego limit theorem.

Prediction theory in Hilbert Spaces: Operator-valued Szego Theory

TL;DR

This work extends Szegő theory to operator-valued orthogonal polynomials on the unit circle in infinite-dimensional settings, enriching prediction theory for wide-sense stationary and nonstationary processes. It introduces an operator-valued OPUC framework via moments, Schur complements, and a Schur-structured Verblunsky recursion, then develops operator versions of Christoffel-Darboux kernels and Bernstein-Szegő approximations. The main result is an operator Szegő limit theorem relating Toeplitz-determinant regularizations to the product over Verblunsky coefficients, with a Wiener-algebra regularity condition ensuring the logarithmic determinant integral is well-defined. The findings generalize matrix-valued theory and provide a robust toolkit for analyzing spectral and predictive properties of large-scale or continuous-time operator-valued processes. \displaystyle \lim_{n\to\infty} \mathrm{det}_{2}(T^{R}_{n})/\mathrm{det}_{2}(T^{R}_{n-1}) = \mathrm{det}\prod_{k=0}^{\infty} (I - \alpha_k \alpha_k^{\dagger}) = \exp\left( \frac{1}{2\pi} \int_{-\pi}^{\pi} \log \det \mu'(\theta) \, d\theta \right)

Abstract

In this paper, we extend some classical results of the Szego theory of orthogonal polynomials on the unit circle to the infinite-dimensional case, and we establish the corresponding Szego limit theorem.
Paper Structure (11 sections, 13 theorems, 104 equations)

This paper contains 11 sections, 13 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Organization of the Paper
  3. Preliminaries
  4. Operator-Valued Measures
  5. Spectral Representation and Stationary Processes
  6. Kolmogorov Isomorphism and Prediction Theory
  7. Operator orthogonal polynomials
  8. Verblunsky recursion
  9. Christoffel-Darboux Formulae
  10. Bernstein-Szegő Approximation
  11. The Szegő Limit Theorem

Key Result

Theorem 2.1

[Kolmogorov decomposition] Let $C$ be a positive definite Toeplitz kernel. Then there exist a Hilbert space $\mathcal{K}$ and a collection of bounded operators $V_n \in \mathcal{L}(\mathcal{H}_n, \mathcal{K})$ such that:

Theorems & Definitions (25)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Remark 3.1
  • Proposition 4.1
  • proof
  • Theorem 4.1: Szegő theorem
  • proof
  • Proposition 4.2
  • ...and 15 more