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Liftings and invariant subspaces of Hankel operators

Sneha B, Neeru Bala, Samir Panja, Jaydeb Sarkar

TL;DR

The paper develops a Hankel-variant lifting framework for intertwiners on model spaces, showing that the intertwining relation $S_u^*X=XS_u$ lifts to a Hankel operator via $X = H_φ|_{Q_u}$ with preserved norm. It then fully classifies Beurling-type invariant and reducing subspaces for Hankel operators in terms of gcd data of inner functions, linking invariance to $P_+φ$ belonging to certain model subspaces. A nonzero intertwiner exists precisely when gcd{u, Jū} ≠ 1, with explicit constructions $X = H_ϕ|_{Q_u}$ obtained from ϕ = T_z^* gcd{u, Jū}$; Blaschke product criteria and Hilbert's Hankel matrix are discussed as concrete illustrations. The concluding remarks relate the results to Toeplitz formulations, dilation theory, and kernel structure, showing how inner functions arise from Hankel kernels via ker H_ϕ = w_ϕ H^2 and ker H_{z̄ Jū} = uH^2.

Abstract

We prove a Hankel-variant commutant lifting theorem. This also uncovers the complete structure of the Beurling-type reducing and invariant subspaces of Hankel operators. Kernel spaces of Hankel operators play a key role in the analysis.

Liftings and invariant subspaces of Hankel operators

TL;DR

The paper develops a Hankel-variant lifting framework for intertwiners on model spaces, showing that the intertwining relation lifts to a Hankel operator via with preserved norm. It then fully classifies Beurling-type invariant and reducing subspaces for Hankel operators in terms of gcd data of inner functions, linking invariance to belonging to certain model subspaces. A nonzero intertwiner exists precisely when gcd{u, Jū} ≠ 1, with explicit constructions obtained from ϕ = T_z^* gcd{u, Jū}$; Blaschke product criteria and Hilbert's Hankel matrix are discussed as concrete illustrations. The concluding remarks relate the results to Toeplitz formulations, dilation theory, and kernel structure, showing how inner functions arise from Hankel kernels via ker H_ϕ = w_ϕ H^2 and ker H_{z̄ Jū} = uH^2.

Abstract

We prove a Hankel-variant commutant lifting theorem. This also uncovers the complete structure of the Beurling-type reducing and invariant subspaces of Hankel operators. Kernel spaces of Hankel operators play a key role in the analysis.
Paper Structure (8 sections, 15 theorems, 137 equations)

This paper contains 8 sections, 15 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. A lifting theorem
  3. Invariant and reducing subspaces
  4. Revisiting liftings and examples
  5. Concluding Remarks
  6. Toeplitz formulation
  7. Dilation formulation
  8. Hankel kernels

Key Result

Theorem 1.2

Let $u \in H^\infty(\mathbb{D})$ be an inner function and let $X\in\mathcal{B}(\mathcal{Q}_u)$. Then $S_u^*X=XS_u$ if and only if there exists $\varphi\in L^{\infty}(\mathbb{T})$ such that In particular, if $S_u^*X=XS_u$, then $X=H_{\varphi}|_{\mathcal{Q}_u}$, and $\|H_\varphi\|=\|X\|$.

Theorems & Definitions (21)

  • Theorem 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Corollary 3.1
  • Corollary 3.2
  • ...and 11 more