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Cyclicity of Cowen-Douglas tuples

Jing Xu, Shanshan Ji, Yufang Xie, Kui Ji

TL;DR

The paper addresses the cyclicity of Cowen-Douglas tuples in $oldsymbol{eta}_n^m(oldsymbol{ abla})$ by integrating operator theory with complex geometry through the associated Hermitian holomorphic vector bundle $E_{oldsymbol{T}}$. The authors develop a constructive approach: for $oldsymbol{T}$ in $oldsymbol{eta}_1^m(oldsymbol{ abla})$ they build a cyclic vector from a holomorphic frame $oldsymbol{eta}(z)= extstyle abla a_oldsymbol{eta} z^oldsymbol{eta}$ and a rapidly decaying coefficient sequence, proving $oldsymbol{H}= ext{span}igl\{oldsymbol{T}^oldsymbol{eta}figrigr\}$. They then extend the result to general $n$ using spanning holomorphic cross-sections guaranteed by Eschmeier–Schmitt (2014), thereby showing that all commuting $m$-tuples in $oldsymbol{eta}_n^m(oldsymbol{ abla})$ are cyclic. This work deepens the link between holomorphic bundle theory and operator cyclicity, generalizing prior single-operator results to the Cowen–Douglas tuple setting with a complete cyclicity characterization.

Abstract

The study of Cowen-Douglas operators involves not only operator-theoretic tools but also complex geometry on holomorphic vector bundles. By leveraging the properties of holomorphic vector bundles, this paper investigates the cyclicity of Cowen-Douglas tuples and demonstrates conclusively that every such tuple is cyclic.

Cyclicity of Cowen-Douglas tuples

TL;DR

The paper addresses the cyclicity of Cowen-Douglas tuples in by integrating operator theory with complex geometry through the associated Hermitian holomorphic vector bundle . The authors develop a constructive approach: for in they build a cyclic vector from a holomorphic frame and a rapidly decaying coefficient sequence, proving . They then extend the result to general using spanning holomorphic cross-sections guaranteed by Eschmeier–Schmitt (2014), thereby showing that all commuting -tuples in are cyclic. This work deepens the link between holomorphic bundle theory and operator cyclicity, generalizing prior single-operator results to the Cowen–Douglas tuple setting with a complete cyclicity characterization.

Abstract

The study of Cowen-Douglas operators involves not only operator-theoretic tools but also complex geometry on holomorphic vector bundles. By leveraging the properties of holomorphic vector bundles, this paper investigates the cyclicity of Cowen-Douglas tuples and demonstrates conclusively that every such tuple is cyclic.
Paper Structure (2 sections, 5 theorems, 35 equations)

This paper contains 2 sections, 5 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Cyclicity of the class $\mathcal{B}_{n}^{m}(\Omega)$

Key Result

Theorem 1.2

ES2014 Let $\Omega\subset\mathbb{C}^{m}$ be a domain of holomorphy, $\mathbf{T}=(T_{1},\cdots,T_{m})\in\mathbf{\mathcal{B}}_{n}^{m}(\Omega)$, and let $\{\gamma_{1},\cdots,\gamma_{n}\}$ be a holomorphic frame of $E_\mathbf{T}$. Then there exist holomorphic functions $\phi_{1},\cdots,\phi_{n}$ such th

Theorems & Definitions (10)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof