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A geometric approach to the compressed shift operator on the Hardy space over the bidisk

Yufeng Lu, Yixin Yang, Chao Zu

TL;DR

The work develops a geometric framework for the compressed shift $S_z$ on $H^2(\mathbb{D}^2)$, using Beurling-type and polynomial quotient modules to illuminate spectral and subspace structure. It introduces generalized Cowen-Douglas operators and ties operator reducibility to the reducibility of associated kernel-space bundles, yielding precise criteria for when $S_z^*$ belongs to the Cowen-Douglas class and when $S_z$ is reducible. The authors provide spectral descriptions, explicit kernel-structure formulas, and reducing-subspace decompositions for rational inner and polynomial quotient settings, including the canonical case $p=z^m-w^n$ and $(z-w)^n$. This geometric-operator-theoretic approach clarifies how zero-set projections govern both spectrum and reducibility, with concrete consequences for the von Neumann algebra $V^*(S_z)$ and for polynomial quotient modules. The results have implications for model-theoretic perspectives on multivariable Hardy spaces and for understanding invariant and reducing subspaces in structured operator contexts.

Abstract

This paper studies the compressed shift operator $S_z$ on the Hardy space over the bidisk via the geometric approach. We calculate the spectrum and essential spectrum of $S_z$ on the Beurling type quotient modules induced by rational inner functions, and give a complete characterization for $S_z^*$ to be a Cowen-Douglas operator. Then we extend the concept of Cowen-Douglas operator to be the generalized Cowen-Douglas operator, and show that $S_z^*$ is a generalized Cowen-Douglas operator. Moreover, we establish the connection between the reducibility of the Hermitian holomorphic vector bundle induced by kernel spaces and the reducibility of the generalized Cowen-Douglas operator. By using the geometric approach, we study the reducing subspaces of $S_z$ on certain polynomial quotient modules.

A geometric approach to the compressed shift operator on the Hardy space over the bidisk

TL;DR

The work develops a geometric framework for the compressed shift on , using Beurling-type and polynomial quotient modules to illuminate spectral and subspace structure. It introduces generalized Cowen-Douglas operators and ties operator reducibility to the reducibility of associated kernel-space bundles, yielding precise criteria for when belongs to the Cowen-Douglas class and when is reducible. The authors provide spectral descriptions, explicit kernel-structure formulas, and reducing-subspace decompositions for rational inner and polynomial quotient settings, including the canonical case and . This geometric-operator-theoretic approach clarifies how zero-set projections govern both spectrum and reducibility, with concrete consequences for the von Neumann algebra and for polynomial quotient modules. The results have implications for model-theoretic perspectives on multivariable Hardy spaces and for understanding invariant and reducing subspaces in structured operator contexts.

Abstract

This paper studies the compressed shift operator on the Hardy space over the bidisk via the geometric approach. We calculate the spectrum and essential spectrum of on the Beurling type quotient modules induced by rational inner functions, and give a complete characterization for to be a Cowen-Douglas operator. Then we extend the concept of Cowen-Douglas operator to be the generalized Cowen-Douglas operator, and show that is a generalized Cowen-Douglas operator. Moreover, we establish the connection between the reducibility of the Hermitian holomorphic vector bundle induced by kernel spaces and the reducibility of the generalized Cowen-Douglas operator. By using the geometric approach, we study the reducing subspaces of on certain polynomial quotient modules.
Paper Structure (5 sections, 34 theorems, 169 equations)

This paper contains 5 sections, 34 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. The spectrum of the compressed shift $S_z$
  3. Geometric property of the compressed shift operators
  4. The Generalized Cowen-Douglas Operator
  5. Polynomial quotient modules $[z^m-w^n]^\perp$ and $[(z-w)^n]^\perp$

Key Result

Lemma 2.1

If $M$ is a $z$-invariant subspace of $H^2(\mathbb{D}^2)$, then where $\sigma (L)$ is the set of points $\lambda\in \mathbb{D}$ for which the left evaluation operator $L(\lambda)$ is not bounded invertible from $M\ominus zM$ to $H^2_{w},$ together with those $\lambda \in \mathbb{T}$ not lying on any of the open arcs of $\mathbb{T}$ on which $L(\lambda)$ is a uni

Theorems & Definitions (65)

  • Definition 1.3
  • Lemma 2.1: DY, Corollary 2.3
  • Lemma 2.2
  • proof
  • Lemma 2.3: Kne, Lemma 3.5
  • Theorem 2.4
  • proof
  • Lemma 2.5: Rud, Theorem 5.4.2
  • Theorem 2.6
  • proof
  • ...and 55 more